Orientation Tensor Research Articles

Stochastic dissipative Euler’s equations for a free body

Abstract Intrinsic thermal fluctuations within a real solid challenge the rigid body assumption that is central to Euler’s equations for the motion of a free body. Recently, we have introduced a dissipative and stochastic version of Euler’s equations in a thermodynamically consistent way (European Journal of Mechanics – A/Solids 103, 105,184 (2024)). This framework describes the evolution of both orientation and shape of a free body, incorporating internal thermal fluctuations and their concomitant dissipative mechanisms. In the present work, we demonstrate that, in the absence of angular momentum, the theory predicts that the principal axes unit vectors of a body undergo an anisotropic Brownian motion on the unit sphere, with the anisotropy arising from the body’s varying moments of inertia. The resulting equilibrium time correlation function of the principal eigenvectors decays exponentially. This theoretical prediction is confirmed in molecular dynamics simulations of small bodies. The comparison of theory and equilibrium MD simulations allow us to measure the orientational diffusion tensor. We then use this information in the Stochastic Dissipative Euler’s Equations, to describe a non-equilibrium situation of a body spinning around the unstable intermediate axis. The agreement between theory and simulations is excellent, offering a validation of the theoretical framework.

Fracture modeling in composite structures containing orthotropic materials by incorporating strain orthogonal decompositions into phase-field method of interfacial damage

PurposeThe phase-field method of interfacial damage is used to simulate the damage in composite structures containing the brittle orthotropic materials and their interface.Design/methodology/approachIn the brittle fracture modeling, the strain tensor is decomposed into positive and negative parts characterizing tension and compression behaviors. By requiring an elastic energy preserving transformation involving the elastic stiffness tensor, these two strain parts must satisfy the orthogonality condition in the sense that the elastic stiffness tensor responds as a metric. However, most of the recent phase-field methods for brittle fracture do not verify this orthogonality condition. Additionally, to describe the damage in structures with anisotropic phases, recent studies have used multiple phase-field variables, with each preferential orientation represented by a phase-field variable to describe the bulk damage of component materials. This approach increases the complexity of simulation procedure. These disadvantages motivate the present study aimed at enhancing the simulation method.FindingsThe present study improves the phase-field method of interfacial damage by (1) incorporating the strain orthogonality condition into the phase-field method; (2) using only one phase-field variable instead of multiple phase-field variables to simulate damage in component orthotropic phases; and (3) investigating the interaction between interfacial damage and bulk damage as well as the effect of orientation tensor of preferential orientation in each orthotropic phase and the interfacial parameters on crack branching in composite structures.Originality/valueThrough several simulation examples, the present simulation method is proven to be accurate, effective, and helps the simulation process simpler than previous relevant methods.

Group-random algorithm to generate representative volume element models for composites

One of the most commonly used methods for characterizing the mechanical properties of discontinuous fiber reinforced composites (DFRC) is to establish a Representative Volume Element (RVE) model and perform finite element (FE) analysis. However, FE analysis on RVE models established by traditional sampling algorithms is often computationally expensive due to the large size of RVE that is required to be statistically representative of the composite. To address this issue, this paper proposes a new approach for constructing RVE models with more accurate description of fiber orientation, aiming to make the FE modelling more efficient by using an RVE with small size. When establishing RVE models with given target fiber orientation tensor, it is very challenging to accurately capture the orientation of fibers. In order to mitigate the error between the orientation tensor reconstructed by fibers generated in the RVE and the target orientation tensor, a group-random algorithm is proposed in the current work to generate RVE models. Unlike the traditional algorithm, in which fibers are sampled one by one in the RVE, the group-random algorithm samples a group of four fibers at one time in order to eliminate the error of the off-diagonal components of the reconstructed orientation tensor in the principal coordinate system. Then a modification tensor is further introduced to mitigate the error of the diagonal components of the reconstructed orientation tensor. Simulation results show that the orientation tensor error could be significantly reduced by the group-random algorithm even for the RVE with low number of fibers. The merits of the group-random algorithm are also witnessed by the stability and accuracy of predicting the elastic constants of composite materials through RVE modeling. It is thus concluded that the major advantage of this work is to provide an alternatively feasible strategy to substantially improve computational efficiency of RVE modelling.

The use of digital thread for reconstruction of local fiber orientation in a compression molded pin bracket via deep learning

A deep convolutional neural network (DCNN) was used for microstructure reconstruction using artificial intelligence (MR-AI) by predicting local average fiber orientation distributions (FOD) in a 3D prepreg platelet molded composite (PPMC) pin bracket. To train the MR-AI model, surface strain fields from residual stresses simulated in PPMC plates were used as the input to the DCNN. A training dataset included PPMC plates with various degrees of global fiber alignment, based on the information obtained from high-fidelity flow simulation of a pin bracket. The MR-AI model was then deployed to analyze FOD in the 3D pin bracket by conducting thermo-elastic residual stress analysis. Initially, the MR-AI model was established entirely on the synthetic simulation data. Then, a μCT scan of a physically molded pin bracket was used to create a finite element model that provided data for additional validation of the DCNN model. For the μCT scan finite element pin bracket the MR-AI model predicted the distribution of fiber orientation tensor components with MAE of 0.10 indicating a global prediction error of 10 %. For the flow simulated pin bracket, the MR-AI model predicted the distribution of fiber orientation tensor components with a global prediction error of 11 %. The MR-AI model showed the ability to predict regions of varying alignment in the base and flange of the pin bracket. The proposed MR-AI methodology allows for rapid prediction of FOD in geometrically complex parts and offers a promising path to detecting unique fiber orientation states in molded components.

Multi-Objectives Optimization of Plastic Injection Molding Process Parameters Based on Numerical DNN-GA-MCS Strategy.

An intelligent optimization technique has been presented to enhance the multiple structural performance of PA6-20CF carbon fiber-reinforced polymer (CFRP) plastic injection molding (PIM) products. This approach integrates a deep neural network (DNN), Non-dominated Sorting Genetic Algorithm II (NSGA-II), and Monte Carlo simulation (MCS), collectively referred to as the DNN-GA-MCS strategy. The main objective is to ascertain complex process parameters while elucidating the intrinsic relationships between processing methods and material properties. To realize this, a numerical study on the PIM structural performance of an automotive front engine hood panel was conducted, considering fiber orientation tensor (FOT), warpage, and equivalent plastic strain (PEEQ). The mold temperature, melt temperature, packing pressure, packing time, injection time, cooling temperature, and cooling time were employed as design variables. Subsequently, multiple objective optimizations of the molding process parameters were employed by GA. The utilization of Z-score normalization metrics provided a robust framework for evaluating the comprehensive objective function. The numerical target response in PIM is extremely intricate, but the stability offered by the DNN-GA-MCS strategy ensures precision for accurate results. The enhancement effect of global and local multi-objectives on the molded polymer-metal hybrid (PMH) front hood panel was verified, and the numerical results showed that this strategy can quickly and accurately select the optimal process parameter settings. Compared with the training set mean value, the objectives were increased by 8.63%, 6.61%, and 9.75%, respectively. Compared to the full AA 5083 hood panel scenario, our design reduces weight by 16.67%, and achievements of 92.54%, 93.75%, and 106.85% were obtained in lateral, longitudinal, and torsional strain energy, respectively. In summary, our proposed methodology demonstrates considerable potential in improving the, highlighting its significant impact on the optimization of structural performance.

Functional description of fiber orientation in paperboard based on orientation tensors resulting from μ-CT scans

Modern μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upmu$$\\end{document}-CT scans offer non-destructive three-dimensional microstructure analysis of various materials. Despite small density contrasts, this technology also accurately captures the complex network structure of paper and paperboard. It provides detailed insight into the fiber orientation distribution in paper, which was previously unattainable with existing measurement methods. The image analysis results are applicable for numerical fiber network models, and simulations based on these models can significantly improve our understanding of the microstructural influence on macrostructural paper properties, such as strength, stiffness, and curl tendency. This work presents a new method for investigating the microstructural fiber orientation in paperboard. Orientation tensors obtained from μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upmu$$\\end{document}-CT scan image processing are extracted and evaluated. Based on the direction of the orientation tensor’s first eigenvector, the fiber orientation distribution is determined and then approximated by periodic and non-periodic probability density functions (PDFs). In this study, 40 commercial paperboard samples, each consisting of three plies, were analyzed. From a given set of commonly used PDFs, the best fitting ones have been identified based on their original location in the paper roll and on their ply. It was found that the von Mises PDF provided the most accurate representation of fiber orientation distribution in the middle ply, while the Elliptical PDF was the most suitable for the outer plies. Moreover, a more pronounced fiber orientation anisotropy was observed at the edges of the paper roll than in its center. The best PDFs and their function parameters are provided to allow for direct usage in numerical microstructure models of paperboard, thus enhancing their representativeness.

White Matter Abnormalities Track Disease Progression in Multiple System Atrophy.

White matter (WM) abnormalities have been implicated in clinically relevant functional decline in multiple system atrophy (MSA). To identify the WM and gray matter (GM) abnormalities in MSA and assess the utility of longitudinal structural and diffusion changes as surrogate markers for tracking disease progression in MSA. Twenty-seven participants with early MSA [15 with clinically predominant cerebellar (MSA-C) and 12 with clinically predominant parkinsonian features (MSA-P)] and 14 controls were enrolled as a part of our prospective, longitudinal study of synucleinopathies. Using structural magnetic resonance imaging (MRI) and diffusion MRI (diffusion tensor and neurite orientation and dispersion density imaging), we analyzed whole and regional brain changes in these participants. We also evaluated temporal imaging trajectories based on up to three annual follow-up scans and assessed the impact of baseline diagnosis on these imaging biomarkers using mixed-effect models. MSA patients exhibited more widespread WM changes than GM, particularly in the cerebellum and brainstem, with greater severity in MSA-C. Structural and diffusion measures in the cerebellum WM and brainstem deteriorated with disease progression. Rates of progression of these abnormalities were similar in both MSA subtypes, reflecting increasing overlap of clinical features over time. WM abnormalities are core features of MSA disease progression and advance at similar rates in clinical MSA subtypes. Multimodal MRI imaging reveals novel insights into the distribution and pattern of brain abnormalities and their progression in MSA. Selected structural and diffusion measures may be useful for tracking disease progression in MSA clinical trials.

Experimental and homogenized orientation-dependent properties of hybrid long fiber-reinforced thermoplastics

This research presents a investigation of long fiber-reinforced thermoplastics (LFT) with mixed fiber types, combining experimental analysis with numerical modeling techniques. By accurately predicting the stiffness of mixed fiber composites, the design margin between mono fiber reinforced materials can be effectively exploited, facilitating the use of such materials. For this purpose in particular, a novel application of the Mori–Tanaka approach with two different inclusions guaranteeing symmetric stiffnesses is presented. This is a method that has never been used before in field studies. In addition, the study integrates fourth-order plate-averaged orientation tensors measured and subsequently interpolated to improve the accuracy of the modeling. Consistency with the established shear-lag modified Halpin–Tsai method is demonstrated, confirming the suitability of both approaches for predicting the tensile modulus of GFLFT and CF+GFLFT. However, discrepancies between predictions and experiments for CFLFT are attributed to the complex microstructure of the material caused by bundling and poor dispersion of the CF. Furthermore, the study reveals remarkable hybridization effects within the mixed fiber LFT, particularly evident in the 22% increase in elongation at break observed in CF+GFLFT compared to CFLFT. Overall, this research significantly advances the understanding and predictive capabilities regarding mixed fiber LFTs, which opens up a new design space of specific properties. This provides valuable insight for future research and industrial applications.

Influence of reconstruction uncertainty in fiber orientation distribution on the effective elastic constants of composites reinforced with discontinuous fibers: a Monte Carlo simulation study

The orientation distribution of fibers in discontinuous fiber composite materials is influenced by various factors associated with the manufacturing process. Predicting fiber orientation distribution can be achieved through software simulation of the process or experimental methods, such as X-ray computed tomography. As uncertainties related to the reconstruction of fiber orientation distribution may be unavoidable in practical cases, this paper investigates the impact of some of these uncertainties on the effective elastic constants of composites through Monte Carlo simulations. The primary objective of this study is to address how the ratio of fiber detection and the measurement error of individual fiber orientation influence the orientation tensors and effective elastic constants. To predict the elastic properties of composites under various scenarios of fiber orientation distributions, a micromechanical model incorporating an orientation averaging procedure has been used. Three cases of different fiber orientation distributions have been analyzed. The conducted Monte Carlo simulations enabled the presentation of a quantitative description of the uncertainty associated with the reconstruction of the orientation distribution of fibers, including the effective elastic constants. The resulting distributions of orientation tensors and elastic constants have been analyzed and discussed.

Comparison of coarse-grained models for cis-1,4-polyisoprene-graphite system

ABSTRACT Structural and dynamical properties of cis-1,4-polyisoprene-graphite system were compared by four coarse-grained models. Coarse-graining of graphene layers replaced four carbon atoms by a single bead while four mapping schemes were used for the polymer. Pressure-correction was applied to the nonbonded part of the coarse-grained potential among coarse-grained polyisoprene beads. Among structural properties density profiles, bond orientation and conformation tensor as function of distance from the graphite surface were studied. Bulk values were achieved at around 2 nm from the graphite surface. Higher degree of coarse-graining showed comparatively larger fluctuations in data. Among dynamical properties mean square displacement and autocorrelation of end-to-end unit vectors for varied distances from the graphite surface were studied. Dynamics study indicated that pressure-correction method used to optimise pressure of the system resulted in slower dynamics for the higher degree of coarse-graining.

Quake-DFN: A Software for Simulating Sequences of Induced Earthquakes in a Discrete Fault Network

ABSTRACT We present an earthquake simulator, Quake-DFN, which allows simulating sequences of earthquakes in a 3D discrete fault network governed by rate and state friction. The simulator is quasi-dynamic, with inertial effects being approximated by radiation damping and a lumped mass. The lumped mass term allows for accounting for inertial overshoot and, in addition, makes the computation more effective. Quake-DFN is compared against three publicly available simulation results: (1) the rupture of a planar fault with uniform prestress (SEAS BP5-QD), (2) the propagation of a rupture across a stepover separating two parallel planar faults (RSQSim and FaultMod), and (3) a branch fault system with a secondary fault splaying from a main fault (FaultMod). Examples of injection-induced earthquake simulations are shown for three different fault geometries: (1) a planar fault with a wide range of initial stresses, (2) a branching fault system with varying fault angles and principal stress orientations, and (3) a fault network similar to the one that was activated during the 2011 Prague, Oklahoma, earthquake sequence. The simulations produce realistic earthquake sequences. The time and magnitude of the induced earthquakes observed in these simulations depend on the difference between the initial friction and the residual friction μi−μf, the value of which quantifies the potential for runaway ruptures (ruptures that can extend beyond the zone of stress perturbation due to the injection). The discrete fault simulations show that our simulator correctly accounts for the effect of fault geometry and regional stress tensor orientation and shape. These examples show that Quake-DFN can be used to simulate earthquake sequences and, most importantly, magnitudes, possibly induced or triggered by a fluid injection near a known fault system.

Predicting the fiber orientation of injection molded components and the geometry influence with neural networks

The injection molding simulation of short fiber reinforced plastics (SFRP) is time consuming. However, until now it is necessary for predicting the local fiber orientation, to optimize the molding process and to predict the mechanical behavior of the material. This research presents the capabilities of artificial neural networks (NN) in predicting fiber orientation tensor (FOT) during injection molding processes, with a focus on enhancing computational efficiency compared to traditional simulation methods. Three NN architectures are compared based on simulated injection molded plates, with the goal of predicting the effect of the plate geometry on the local fiber orientation. Results indicate that NN outperform the baseline assumption of aligned fibers and demonstrate significant potential for accurate FOT prediction. The computational efficiency of NN, especially during the prediction phase, showcases a reduction in processing time by a factor of 104 compared to traditional simulation methods. This research lays a foundation for further exploration into the feasibility of NN in partly replacing time-consuming simulations for practical applications in injection molding processes.

Fingerprints of Chalcogen Bonding Revealed Through 77Se-NMR.

77Se-NMR is used to characterise several chalcogen bonded complexes of derivatives of the organoselenium drug ebselen, exploring a range of electron demand. NMR titration experiments support the intuitive understanding that chalcogen bond donors bearing more electron withdrawing substituents give rise stronger chalcogen bonds. The chemical shift of the selenium nucleus is also shown to move upfield as it participates in a chalcogen bond. Solid-state NMR is used to explore chalcogen bonding in co-crystals. Due to the lack of molecular reorientation on the NMR timescale in the solid state, the shape of the chemical shift tensor can be determined using this technique. A range of co-crystals are shown to have extremely large chemical shift anisotropy, which suggests a strongly anisotropic electron density distribution around the selenium atom. A single crystal NMR experiment was conducted using one of the co-crystals, affording the absolute orientation of the chemical shift tensor within the crystal. This showed that the selenium nucleus is strongly shielded in the direction of the chalcogen bond (due to the approach of the lone pair of the Lewis base), and strongly deshielded in the perpendicular direction. The orientation of the deshielded axis is consistent with the presence of a second σ-hole which is not participating in a chalcogen bond, showing the profound effect of electron density anisotropy on the chemical shift.

On the effectiveness of deep material networks for the multi-scale virtual characterization of short fiber-reinforced thermoplastics under highly nonlinear load cases

A key challenge for the virtual characterization of components manufactured using short fiber-reinforced thermoplastics (SFRTs) is the inherent anisotropy which stems from the manufacturing process. To address this, a multi-scale approach is necessary, leveraging deep material networks (DMNs) as a micromechanical surrogate, for a one-stop solution when simulating SFRTs under highly nonlinear long-term load cases like creep and fatigue. Therefore, we extend the a priori fiber orientation tensor interpolation for quasi-static loading (Liu et al. in Intelligent multi-scale simulation based on process-guided composite database. arXiv:2003.09491, 2020; Gajek et al. in Comput Methods Appl Mech Eng 384:113,952, 2021; Meyer et al. in Compos Part B Eng 110,380, 2022) using DMNs with a posteriori approach. We also use the trained DMN framework to simulate the stiffness degradation under fatigue loading with a linear fatigue-damage law for the matrix. We evaluate the effectiveness of the interpolation approach for a variety of load classes using a dedicated fully coupled plasticity and creep model for the polymer matrix. The proposed methodology is validated through comparison with composite experiments, revealing the limitations of the linear fatigue-damage law. Therefore, we introduce a new power-law fatigue-damage model for the matrix in the micro-scale, leveraging the quasi-model-free nature of the DMN, i.e., it models the microstructure independent of the material models attached to the constituents of the microstructure. The DMN framework is shown to effectively extend material models and inversely identify model parameters based on composite experiments for all possible orientation states and variety of material models.

Anisotropic evolution of viscous strain in soft biological materials

We propose a model for anisotropic viscoelastic biological materials that can handle large deformations, based on the kinematic assumption that the reinforcing fibre structure undergoes affine deformation with the underlying matrix. A generalized orientation tensor approach is used to account for the dispersion of the fibres. Moreover, we consider a strain energy function that features both an elastic and an overstress component, corresponding to distinct natural states. As a consequence of this choice, the remodelled state is not necessarily stress-free, and the material does not completely relax the stress. Notably, we consider that viscous remodelling also alters the fibre distribution, leading to a dependence of the overstress energy on the remodelled orientation tensor. An anisotropic evolution equation for the viscous strain is then derived, which has five distinct characteristic times if a single fibre family is considered and requires no additional assumptions on the viscous spin. To implement the model, we prove that the evolution of the viscous strain can be recast in a variational form by an Onsager variational principle. Finally, we discuss the algorithm used for the simulations and show numerical examples that serve as benchmark test cases for viscoelastic materials.

Impact of faults on the remote stress state

Abstract. The impact of faults on the contemporary stress field in the upper crust has been discussed in various studies. Data and models clearly show that there is an effect, but so far, a systematic study quantifying the impact as a function of distance from the fault is lacking. In the absence of data, here we use a series of generic 3-D models to investigate which component of the stress tensor is affected at which distance from the fault. Our study concentrates on the far field, located hundreds of metres from the fault zone. The models assess various techniques to represent faults, different material properties, different boundary conditions, variable orientation, and the fault's size. The study findings indicate that most of the factors tested do not have an influence on either the stress tensor orientation or principal stress magnitudes in the far field beyond 1000 m from the fault. Only in the case of oblique faults with a low static friction coefficient of μ=0.1 can noteworthy stress perturbations be seen up to 2000 m from the fault. However, the changes that we detected are generally small and of the order of lateral stress variability due to rock property variability. Furthermore, only in the first hundreds of metres to the fault are variations large enough to be theoretically detected by borehole-based stress data when considering their inherent uncertainties. This finding agrees with robust stress magnitude measurements and stress orientation data. Thus, in areas where high-quality and high-resolution data show gradual and continuous stress tensor rotations of >20∘ observed over lateral spatial scales of 10 km or more, we infer that these rotations cannot be attributed to faults. We hypothesize that most stress orientation changes attributed to faults may originate from different sources such as density and strength contrasts.

Topology Optimization and Efficiency Evaluation of Short-Fiber-Reinforced Composite Structures Considering Anisotropy

The current study aims to develop a methodology for obtaining topology-optimal structures made of short fiber-reinforced polymers. Each iteration of topology optimization involves two consecutive steps: the first is a simulation of the injection molding process for obtaining the fiber orientation tensor, and the second is a structural analysis with anisotropic material properties. Accounting for the molding process during the internal iterations of topology optimization makes it possible to enhance the weight efficiency of structures—a crucial aspect, especially in aerospace. Anisotropy is considered through the fiber orientation tensor, which is modeled by solving the plastic molding equations for non-Newtonian fluids and then introduced as a variable in the stiffness matrix during the structural analysis. Structural analysis using a linear anisotropic material model was employed within the topology optimization. For verification, a non-linear elasto-plastic material model was used based on an exponential-and-linear hardening law. The evaluation of weight efficiency in structures composed of short-reinforced composite materials using a dimensionless criterion is addressed. Experimental verification was performed to confirm the validity of the developed methodology. The evidence illustrates that considering anisotropy leads to stiffer structures, and structural elements should be oriented in the direction of maximal stiffness. The load-carrying factor is expressed in terms of failure criteria. The presented multidisciplinary methodology can be used to improve the quality of the design of structures made of short fiber-reinforced composites (SFRC), where high stiffness, high strength, and minimum mass are the primary required structural characteristics.

On the Averaging and Closure of Fiber Orientation Tensors in Virtual Process Chains

Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space.

An equivalent anisotropy orientation tensor algorithm for integrated material-structure design

This study presents the Equivalent Anisotropy Orientation Tensor (EAOT) algorithm as a novel approach for the integrated design of thin-walled structures composed of composite materials. The EAOT algorithm is designed to enhance the buckling load of structures by efficiently optimizing the distribution of strain energy density. In addition, the algorithm considers the fiber orientation and stiffener structure as orthogonal isotropic material, providing an innovative design methodology that surpasses conventional approaches. The EAOT algorithm incorporates a weighted function to account for the effects of different buckling modes, which solves the problem of mode overlap and exchange that affects the optimization effect. Utilizing the optimization results from the EAOT algorithm enables the realization of effective material-structure integrated design. Two case studies demonstrate the superior performance of designs based on this algorithm compared to conventional methods, resulting in a substantial increase in the critical buckling load by 15.3 % to 34.7 %.

On the influence of the fourth order orientation tensor on the dynamics of the second order one

Abstract The consequences of introducing the fourth order orientation tensor as an independent variable in addition to the second order one are investigated. In the first part consequences of the Second Law of Thermodynamics are exploited. The cases with the second order alignment tensor in the state space on one hand and with the second and fourth order alignment tensors on the other hand are analogous. In the latter case differential equations for the second and fourth order tensors result from linear force-flux relations with a coupling arising due to coupling terms in the free energy. In the second part the differential equations for the second order orientation tensor or the second and fourth order orientation tensors, respectively are given explicitly in the special case of a rotation symmetric orientation distribution. The Folgar-Tucker equation with a quadratic closure relation leads to a Riccati equation for the second order parameter. In comparison the Folgar-Tucker equation and the differential equation for the fourth order parameter are considered. The fourth order parameter is eliminated later. The resulting equation for the second order parameter is a Duffing equation with a behavior of solutions completely different from the solutions of the Riccati equation.