Multiplicative Decomposition Research Articles

Micro-Slip-Induced Multiplicative Plasticity: Existence of Energy Minimizers

To account for material slips at microscopic scale, we take deformation mappings as SBV functions varphi , which are orientation-preserving outside a jump set taken to be two-dimensional and rectifiable. For their distributional derivative F=Dvarphi we examine the common multiplicative decomposition F=F^{e}F^{p} into so-called elastic and plastic factors, the latter a measure. Then, we consider a polyconvex energy with respect to F^{e}, augmented by the measure |textrm{curl},F^{p}|. For this type of energy we prove the existence of minimizers in the space of SBV maps. We avoid self-penetration of matter. Our analysis rests on a representation of the slip system in terms of currents (in the sense of geometric measure theory) with both mathbb {Z}^{3} and mathbb {R}^{3} valued multiplicity. The two choices make sense at different spatial scales; they describe separate but not alternative modeling options. The first one is particularly significant for periodic crystalline materials at a lattice level. The latter covers a more general setting and requires to account for an energy extra term involving the slip boundary size. We include a generalized (and weak) tangency condition; the resulting setting embraces gliding and cross-slip mechanisms. The possible highly articulate structure of the jump set allows one to consider also the resulting setting even as an approximation of climbing mechanisms.

PENERAPAN METODE MULTIPLICATIVE DECOMPOSITION DALAM MEMPREDIKSI JUMLAH KASUS DBD DI RSU. HAJI MEDAN

DHF case are diseases caused by four types of dengue virus, where each type has its own characteristics and pattern of spread. The purpose of this study is to make predictions on DHF cases in the future. This study uses the Multiplicative Decomposition method, in which this method breaks down time series data into four patterns, namely trend, seasonality, cycles and error. This research was conducted at RSU Haji Medan using DHF case data from January 2017 to December 2021. The results showed that the Multiplicative Decomposition method has an MSE value of 0,0012 and a MAPE value of 0,23%, which means it has a very good accuracy value because less than 10%. The conclusion of the study is that the Multiplicative Decomposition method is a method in the good category in predicting future DHF cases

Actin based motility unveiled: How chemical energy is converted into motion

Actin-based motility is a complex process in which the actin-polymerization is the primary force-generating motor machinery. It can produce protrusive forces through actin filaments polymerization and cross-link during lamellipodia protrusion in migrating cells and it is responsible for the intracellular motion of certain pathogens in infected host cells. We propose a multi-physics model for actin-based motility, stemming from continuity equations that account for the actin chemical kinetics. Thermodynamic restrictions are identified, moving from the multiplicative decomposition of the deformation gradient into chemical and elastic parts. Constitutive theory and chemical kinetics are prescribed to finally write governing equations for the multi-physics problem. The field equations are solved numerically with the finite element method. As a proof of concept, a one-dimensional model for actin-based motility of bacteria pathogens is studied.

Thermo-elasticity displacement formulation for constrained articulated mechanical systems

This paper proposes an approach for thermal analysis of articulated systems subject to boundary and motion constraints (BMC). The solution framework is designed to capture large temperature fluctuations, significant change in geometry due to reference configuration and deformations, and geometric nonlinearity due to articulated mechanical systems (AMS) large displacements and spinning motion. Thermal-expansion displacements, which do not contribute to rigid-body translations, are determined from thermal stretch of position-gradient vectors using a new sweeping matrix technique designed to eliminate dependence on translational rigid-body modes. A new quadratic thermal-energy kinetic form is defined and used to formulate a dynamic force vector that accounts for thermal transient and inertia effects. Nodal thermal displacements are used in formulating AMS differential/algebraic equations (DAEs), and consequently, thermal stresses due to BMC equations are automatically accounted for based on integration of thermal analysis and Lagrange-D’Alembert principle, which is the foundation of computational multibody system (MBS) algorithms. The approach used in this study for large-displacement constrained and unconstrained thermal expansions is based on multiplicative decomposition of position-gradient matrix, instead of strain additive decomposition, for solution of thermo-elasticity problems. Four configurations are used to define continuum geometry and displacements: straight configuration, reference configuration, thermal-expansion configuration, and current configuration. The proposed approach allows applying thermal loads during constrained large displacements, does not impose restrictions on the choice of thermal coefficients, captures reference-configuration geometry and change in inertia forces due to temperature fluctuations, accounts for thermal displacement in formulating AMS nonlinear constraint equations, and allows for integration with MBS algorithms for the study of a wide range of thermo-elasticity problems.

A multiplicative finite strain deformation for diffusion-induced stress: An incremental approach

Understanding the evolution of lithium concentration and stresses in electrode materials during electrochemical cycling is of paramount importance in studying the fundamental mechanisms contributing to the structural degradation of lithiated electrode materials. In this work, we use the change rate of the volume of a host material in analyzing diffusion-induced volume expansion and obtain the volume expansion as an exponential function of the concentration of solute atoms and the coefficient of linear expansion in contrast to the widely used linear relation between the diffusion-induced volume expansion and the coefficient of linear expansion. Using the principle of multiplicative decomposition in the framework of elastic deformation and thermodynamic free energies, we derive the constitutive relation for diffusion-induced finite deformation of elastic materials. The symmetric Piola-Kirchhoff stress is an exponential function of the coefficient of linear expansion and the concentration of solute atoms. Incorporating the change of strain field with the change in the number of solute atoms in Helmholtz free energy, we derive the chemical potential of the solute atoms in the host material the same as the one given by Li et al. (Zeitschrift für PhysikalischeChemie, 49 (1966) 271–290). Numerical results for the lithiation of a spherical silicon electrode reveal that the spatial distributions of lithium concentration and stresses for small dimensionless time (dilute solution) exhibit similar trends to the corresponding ones calculated from the theory of linear elasticity and nonlinear characteristics for large dimensionless time (concentrated solution).

Two sides of the same coin: The economic and environmental effects of China's international trade from a global value chain perspective

China's accession to the World Trade Organization (WTO) has led to rapid economic growth and international trade development. However, China is also challenged with a heavy environmental burden due to the massive carbon emissions transferred through trade. By splitting production activities into traditional trade and global value chain (GVC) activities, this paper uses an intercountry input-output (ICIO) framework to study the imbalances of the economic and environmental effects between China's imports and exports at different levels. We define the indices value added per embodied emission in imports and exports (VPM and VPX, respectively). Additionally, we find a large gap between China's VPM and VPX, primarily because developed economies gain much higher value added per embodied emission than China gained through exports in GVC activities. Then, we study how to narrow the gap between China's VPM and VPX. The application of multiplicative structural decomposition analysis (SDA) and the logarithmic mean Divisia index I (LMDI-I) approach reveals the total and bilateral-sectoral contributions of the driving factor effects to the changes in China's VPM and VPX. The results provide tailored implications for promoting the comprehensive economic and environmental benefits of China's imports and exports.

A novel anisotropic stress‐driven model for bioengineered tissues accounting for remodeling and reorientation based on homeostatic surfaces

AbstractA co‐rotated formulation of the intermediate configuration is derived in a thermodynamically consistent manner. As a result of this formulation, algorithmic differentiation (AD) and the equations of the material model can be combined directly, i.e., the equations can be implemented into the AD tool and the corresponding derivatives can be calculated using AD. This is not possible when the equations are given in terms of the intermediate configuration, since the multiplicative decomposition suffers from an inherent rotational non‐uniqueness. Moreover, a novel stress‐driven kinematic growth model is presented that takes homeostasis and fiber reorientation into account and is based on the co‐rotated formulation. A numerical example reveals the promising potential of both the co‐rotated formulation and the stress‐driven growth model.

On the time-dependent mechanics of membranes via the nonlinear finite element method

In this work, the problem of finite generalized and viscoelastic deformation of thin membranes with different geometries, made of incompressible hyperelastic materials, is formulated. The multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts, and making use of dissipation inequality, nonlinear evolution equations for the internal variables of the models are obtained. The mechanical behavior of the dampers is assumed to be linearly viscous. Therefore, the Cauchy-like stress in the dampers is similar to that in Newtonian fluids and includes terms for the hydrostatic pressure and viscosity. The implicit and second-order accurate trapezoidal method is employed for the time integration of the evolution equations. Due to the highly nonlinear governing differential equations including the effects of geometric nonlinearity and viscoelasticity, a nonlinear finite element formulation based on isoparametric elements is developed. The accuracy and performance of the developed formulation and time-dependent solutions are verified by studying several numerical examples. The obtained results are compared with theoretical and experimental data available in the literature. The proposed formulation can appropriately predict the experimental results of viscoelastic membranes for both in-plane and out-of-plane deformations.

Three-dimensional growth simulation of swellable soft materials based on CS-FEM

In this paper, a three-dimensional numerical framework for modeling growth of swellable soft materials at large deformation is established based on the cell-based smooth finite element method, and the multiplicative decomposition scheme of deformation gradient is given. The second P-K stress and Green's strain tensor are selected as work conjugate pairs, and the corresponding mathematical expressions of stiffness matrix and geometric stiffness matrix are derived. The numerical method is implemented based on Matlab platform, and the isotropic and anisotropic growth behaviors of swellable soft materials are simulated respectively. The results show that anisotropic growth will inhibit the deformation of expandable soft materials compared with isotropic growth. The simulation results are compared with the calculation results in the existing literature. The comparison results show that the characteristics and the morphological mode are in good agreement, which proves the effectiveness of the numerical framework in simulating the growth behavior of expandable soft materials at large deformation, and is able to reveal the mechanical mechanism of the plant growth phenomenon in nature.

Field dependent magneto-viscoelasticity in particle reinforced elastomer

Magneto-rheological elastomers (MREs) are intelligent polymers that contain magneto-active particles in an elastomeric polymer matrix. The introduction of active particles influences MREs’ stiffness and rheological behavior under magneto-mechanical loading. The present study aims to construct a micro-mechanics-based constitutive model for MREs that accounts for the active particle volume fraction. The theory is extended in the finite deformation, and a multiplicative decomposition of the deformation gradient into elastic and viscous components is adopted, yielding a transitional configuration on which the magnetic field vectors may be transported. The current study explores micro-mechanics perspective of material behavior in finite deformation. The magnetic field vectors are implicitly revealed to have equilibrium and non-equilibrium components. This indicates that both mechanical viscoelastic effects and the material’s resistance to magnetization significantly contribute to mechanical dissipation. The second law of thermodynamics is used to develop a constitutive relation for finite deformation in magneto-viscoelastic material. The proposed material model is validated with the experimental data using the least possible material parameters.

ANALISIS PERAMALAN PENJUALAN DAN KEUNTUNGAN SAYURAN BUNCIS (Studi Kasus Di Pt. Tiga Bintang Sukses Jatiasih Bekasi Jawa Barat)

PT. Tiga Bintang Sukses (PT. TBS) as the holder of the Goedang Mayoer trademark is a company that strives in the field of providing the best fresh vegetables and fruits, by promoting the concept of farm to table or from farmers directly to consumers. One of the vegetables sold is chickpeas. Beans are one of the highest selling vegetables in the company but fluctuate due to uncertain demand from consumers, especially hotels, restaurants, cafes, and offices. In meeting consumer demand, the company cooperates with farmers. The purpose of this study is to find the best forecasting technique for vegetable sales in the company, to make a sales forecast level and to make an average profit from the sales forecast for the next six months for green beans at PT. Three Stars of Success. This study uses a quantitative approach. The method used in this research is a descriptive analysis and case study. Data collection techniques used using primary data and secondary data. Primary data is related to the company and uses a questionnaire. Secondary data comes from the company's internal data, namely sales of green beans from September 2021 to February 2022. The best forecasting technique for green beans is the multiplicative decomposition technique. The results of the level of sales forecasting show that the highest level of sales for green beans is 500.61 kg. The profit forecast for the next 6 months, has an average profit for green beans, which is Rp. 1,596,232.

Internally balanced hyperelastic constitutive model in terms of principal stretches

Internal variables are included in material constitutive models to capture realistic mechanical effects such as viscosity. Recent work for compressible hyperelastic material is developed based on applying the argument of calculus variation to two-factor multiplicative decomposition of the deformation gradient. Strain energy function is expressed in terms of principal invariants of the deformation gradient decomposed counterparts. The present work permits the use of strain energy function in terms principal stretches of the deformation gradient multiplicatively decomposed counterparts directly. This simplifies significantly the expression of second Piola–Kirchhoff stress and internal balance tensor.

On additive and multiplicative decompositions of sets of integers composed from a given set of primes, II (Multiplicative decompositions)

On additive and multiplicative decompositions of sets of integers composed from a given set of primes, II (Multiplicative decompositions)

Rate-dependent stress-order coupling in main-chain liquid crystal elastomers.

Liquid crystal elastomers (LCEs) exhibit significant viscoelasticity. Although the rate-dependent stress-strain relation of LCEs has already been widely observed, the effect of the intricate interplay of director rotation and network extension on the viscoelastic behavior of main-chain LCEs remains inadequately understood. In this study, we report real-time measurements of the stress, director rotation, and all strain components in main-chain nematic LCEs subjected to uniaxial tension both parallel and tilted to the initial directors at different loading rates and relaxation tests. We find that both network extension and director rotation play roles in viscoelasticity, and the characteristic relaxation time of the network extension is much larger than that of the director rotation. Interestingly, the gradual change of the director in a long-time relaxation indicates the director reorientation delay is not solely due to the viscous rotation of liquid crystals but also arises from its coupling with the highly viscous network. Additionally, significant rate-dependent shear strain occurs in LCEs under uniaxial tension, showing non-monotonic changes when the angle between the stretching and the initial director is large enough. Finally, a viscoelastic constitutive model, only considering the viscosity of the network by introducing multiplicative decomposition of the deformation gradient, is utilized to manifest the relation between rate-dependent macroscopic deformation and microscopic director rotation in LCEs.

Trends of hydroclimatic variables and outflow in the Athabasca River basin

Our study focuses on assessing trends of hydroclimatic variables and their impact on the runoff regime in the Athabasca River basin in two periods (from the beginning of monitoring until 2011 and 1971−2011). Applied methods included the Mann-Kendall statistical test, linear regression, multiplicative decomposition of data series, the Indicators of Hydrologic Alteration, and the Range of Variability Approach. Temperature records indicated a strong regional warming trend, which influenced decreased snowfall and a declining trend in spring runoff. Our results indicate that between 1971 and 2011 median discharge values decreased by > 20% on the lower and middle courses of the Athabasca River. Although on the upper course the median long-term minimum discharge increased due to the melting of mountain glaciers, on the middle and lower courses a significant decrease occurred. The discharge variability has also increased. All factors and changes are important because they affect the Athabasca-Peace delta’s ecosystem and socioeconomic activities on the lower course.

Mechanical modeling of growth applied to Saccharomyces cerevisiae yeast cells

A theoretical and numerical model is developed to describe the growth of Saccharomyces cerevisiae yeasts. This kind of cells is considered here as an axisymmetrical and deformable structure, the inner surface of which is continuously acted upon by a high turgor pressure. Due to the small ratio between the cellwall thickness and the cell radius, a structural shell approach is used. Moreover, the finite strain range is assumed because of the soft nature of these cells. The adopted kinematics is herein based on the multiplicative decomposition of the deformation gradient into an elastic part Fe and an irreversible part related to the growth Fg, i.e. F = FeFg. The reversible response is described using an hyperelastic model of the Ogden type. In accordance with continuum thermodynamics requirements, a criterion is introduced to control the evolution of the growth phenomenon. In this latter two parameters are involved: a growth stress-like threshold, and a growth characteristic time. Embedded within the finite element framework, an illustrative example shows the growth phenomenon of spherical cells going from yeast bud emergence to the step just before cell division. A parametric study highlights the influence of the above mentioned parameters on the cell responses.

Isomorphism Continuum Stored Energy Functional for Finite Thermoelastic Deformation

Continuum mechanics for isotropic finite thermoelastic deformations have been reviewed. Thermal effects on mechanical responses of rubbers have been captured by the isomorphism continuum stored energy (CSE) functional with the multiplicative decomposition of deformation gradient while preserving the structure of symmetry for finite structural deformation. The CSE finite thermoelastic model fits and predicts experimental data of SR and NR-C60 rubbers at different external temperatures. For internal temperature effects of both NR and NR-SIC rubbers, the CSE finite thermoelastic model of stored energy and entropy, along with the newly developed CTE and CI models, fits both nominal stress-stretch and temperature change-stretch experimental data in uniaxial extension tests.

A Homogenized Bending Theory for Prestrained Plates

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature non-flat equilibrium shapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient. By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a Gamma -limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

Inelastic material formulations based on a co-rotated intermediate configuration—Application to bioengineered tissues

In the field of material modeling, there is a general trend to include more and more complex phenomena in the modeling, making the models’ theoretical derivation and numerical implementation extremely difficult. In particular, modeling inelastic material behavior under finite deformations in a continuum mechanical manner, e.g. to simulate soft biological tissues, remains one of the most challenging tasks. Unfortunately, the multiplicative decomposition of the deformation gradient usually utilized in this context suffers from an inherent rotational non-uniqueness, making it not straightforward to combine this approach with algorithmic differentiation (AD)—a very helpful tool in modern computational mechanics. To address this issue in the growth and remodeling model proposed herein, a novel co-rotated intermediate configuration is introduced. This configuration shares essential characteristics with the intermediate one, but is uniquely defined and applicable to a wide range of inelastic materials. In this regard, the concept of structural tensors, hardening effects, and a thermodynamically consistent derivation are discussed as well. Since the stress-driven growth model presented is based on the approach of homeostatic surfaces by Lamm et al. (2022), a large number of derivatives of potentials and energies are required, which can be elegantly implemented using AD due to the co-rotated formulation. Moreover, fiber remodeling of collagen fibers is taken into account in a stress-driven manner using AD. Finally, qualitative comparisons are made with recently published experiments by Eichinger et al. (2020) in uniaxial and multiaxial settings, revealing the efficient combination of the proposed framework and the material model.

Constrained Large-Displacement Thermal Analysis

Abstract Two different cases are encountered in the thermal analysis of solids. In the first case, continua are not subject to boundary and motion constraints and all material points experience same displacement-gradient changes as the result of application of thermal loads. In this case, referred to as unconstrained thermal expansion, the thermal load produces uniform stress-free motion within the continuum. In the second case, point displacements due to boundary and motion constraints are restricted, and therefore, continuum points do not move freely when thermal loads are applied. This second case, referred to as constrained thermal expansion, leads to thermal stresses and its study requires proper identification of the independent coordinates which represent expansion degrees-of-freedom. To have objective evaluation and comparison between the two cases of constrained and unconstrained thermal expansion, the reference-configuration geometry is accurately described using the absolute nodal coordinate formulation (ANCF) finite elements. ANCF position-gradient vectors have unique geometric meanings as tangent to coordinate lines, allowing systematic description of the two different cases of unconstrained and constrained thermal expansions using multiplicative decomposition of the matrix of position-gradient vectors. Furthermore, generality of the approach for large-displacement thermal analysis requires using the Lagrange–D'Alembert principle for proper treatment of algebraic constraint equations. Numerical results are presented to compare two different expansion cases, demonstrate use of the new approach, and verify its results by comparing with conventional finite element (FE) approaches.