Stress Vector Research Articles

Rational design of a 500 m3 fermenter for erythromycin production by Saccharopolyspora erythraea

Erythromycin is a macrolide antibiotic produced by Saccharopolyspora erythraea. Its yield is greatly affected by the fermentation conditions and the bioreactor configurations. In this study, a novel scale-up method for erythromycin fermentation was developed based on computational fluid dynamics (CFD) and time constant analysis. Firstly, the dissolved oxygen (DO) was determined as a key parameter according to the physiological properties of S. erythraea cultivated in a 50 L bioreactor. It was found that the time constant of oxygen supply (tmt) in a 500 m3 bioreactor should be less than 6.25 s in order to satisfy the organism's oxygen uptake rate (OUR). Subsequently, a 500 m3 bioreactor was designed using the time constant method combined with empirical correlations. The impeller combination with one BDT8 impeller at bottom and two MSX4 impellers at upper part was determined, and then validated by numerical simulation. The results indicated that the tmt of the bioreactor (< 6.25 s) and the fluid properties, including gas hold-up, shear stress and fluid vector, met the requirements of erythromycin fermentation. Finally, the industrial production of erythromycin in the 500 m3 showed the design method was applicable in large scale fermentation.

An accelerated method of Uzawa algorithm in contact problems

A lot of methods are applied in the numerical solution of contact problems, among which the most widely used method is known as Uzawa algorithm. A contact problem can transform into a saddle problem. We use Uzawa algorithm for this saddle format. Because this form can get both displacements for nodes and outward normal components of stress vectors for boundary nodes. Although the Uzawa algorithm is easy to implement, its convergent velocity is slow. Taking this algorithm as a fixed point iteration, we can use the Anderson acceleration to improve the convergent velocity.

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of implicit constitutive relations we establish the existence and uniqueness of a global-in-time large-data weak solution. We then focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises, which is that the Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfill natural compatibility conditions.

An Anisotropic Elastic Solid With an Elliptical Inhomogeneity Under a Non-Uniform In-Plane And Anti-Plane Remote Loading

Summary We consider the case of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to a non-uniform remote loading described by remote stresses and strains which are linear functions of the two in-plane coordinates. The internal stresses and strains within the elliptical inhomogeneity are found to be linear functions of the two in-plane coordinates. In addition, we obtain explicit real-form solutions describing the elastic field inside the inhomogeneity as well as hoop stress vectors and hoop stresses on the matrix side and on the inhomogeneity side. We also obtain the corresponding solutions for the two limiting cases in which the elliptical inhomogeneity takes the form of a hole or a rigid inhomogeneity. The solution method presented here can be extended to accommodate the more general scenario in which the remote applied stresses and strains are arbitrary-order polynomials of the two in-plane coordinates.

A Three-Field Variational Formulation for a Frictional Contact Problem with Prescribed Normal Stress

In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational formulation by means of a new variational approach governed by the theory of bipotentials combined with a Lagrange-multipliers technique. In this new approach, the unknown of the mechanical model is a triple consisting of the displacement field, a Lagrange multiplier related to the friction force and the Cauchy stress tensor. We obtain existence, uniqueness, boundedness and convergence results.

Current Stress Pattern and Geodynamics of the Baikal Rift System

Abstract—The crustal stress field of the Baikal Rift System has been reconstructed by tectonophysical inversion of focal mechanisms from the catalog of earthquakes recorded by the regional seismological network. Cataclastic analysis of fault slip data developed at the Shmidt Institute of the Physics of the Earth (Moscow) revealed previously unknown features in the behavior of principal stresses. Namely, the maximum deviatoric stresses diverge off the rift axis while the normalized spherical and deviatoric stress tensor components reach high magnitudes in the crust of the Baikal Basin. The obtained stress pattern of the Baikal Rift System is consistent with the rift origin by a joint action of a vertical mantle flow (upwelling branch of convection) and a horizontal flow in the asthenosphere which drives the NW–SE motion of the Amur plate off Eurasia.

Large deformation nonlinear bending analysis of multilayer functionally graded graphene-reinforced skew microplate under mechanical and thermal loads using FSDT and MCST

ABSTRACT In this paper large deformation of the skew microplate made of laminated composite enhanced with graphene nanosheets is investigated using first shear deformation theory (FSDT) when the plate undergoes combination of mechanical and thermal loads. Halpin-Tsai model is utilised to model mechanical properties while for thermal properties Schapery model is considered. First, shear deformation of the plate was obtained while using Green-Lagrange tensor the non-linear strain of the microplate is extracted using the von-Karman assumptions. Then, by Modified Coupled Stress Theory (MCSD), the components of Cauchy and Couple stress tensors are specified. Finally, using Hamilton Principle the governing equation of the microplate and boundary conditions are computed. After deriving analytical governmental equation results are presented for the simple support boundary condition and the proposed model is validated by previously reported data and it proves to be promising. Primary results showed that with raising skew angle deflection and bending moment decreases which means microplate becomes more rigid in higher skew angles. Higher mechanical loads lead to higher deflections and bending moments. Four different configurations of the layers were analysed from which angle-ply configuration possessed highest deflection and bending while UD configuration revealed the lowest values of deflection and bending moment.

Analysis of displacement path dependence in mixed mode cohesive law

In the present work, a general theoretical analysis is presented and novel mathematical expressions are derived to obtain the cohesive law of adhesive joints under mixed-mode loading.It is theoretically stated that the mixed-mode truss-like cohesive laws, where the stress vector is forced to be parallel to the displacement vector, cannot adequately describe the fracture process when the displacement path is non-linear.Finally, an experimental proof is given to show that in a Mixed Mode Bending test the difference between the phase angles of the stress vector and displacement differential vector must be taken into account when the displacement path becomes nonlinear, so that the cohesive stress will no longer be a function only of the magnitude of the displacement.

A non-linear complementary energy-based constitutive model for incompressible isotropic materials

Inverse non-linear stress–strain relations between the logarithmic Hencky strain tensor and the Cauchy stress tensor are derived for incompressible isotropic materials. The constitutive model is based on a new type of complementary energy potential that considered to be the function of the second and third invariants of the deviatoric Cauchy stress tensor in a power-law form. A proper scaling of the third deviatoric stress invariant allows separate fitting to uniaxial tension and equibiaxial tension (or uniaxial compression) data. The predictive capabilities of a four- and a six-parameter model are presented through a parameter fitting procedure using Treloar’s experimental data for rubber. The performance of the new constitutive model is also demonstrated by solving the inflation problem of a spherical shell.

Difference analysis of phenomenological models with two variable forms for soft tissue quasi-static mechanical characterization

It was important to understand the accurate mechanical properties of soft tissue for the evaluation of its injury, provide reliable protective ways or design effective human resist-injury devices. There was little study that clarified the difference between phenomenological models based on strain invariant and the principal stretches variables respectively although some quasi-static constitutive models of soft tissue were developed. In this study, we enumerate several typical hyperelastic models and derive the tensor equation of stress-strain based on continuum mechanics to fit the experimental data of human brain specimens under multiple loading modes in previous studies and give the coefficient of determination based on the least square fitting. It was suggested that two variable forms of phenomenological models with only the first strain invariant are consistent under uniaxial compression and tension, but the Cauchy stress tensor expressed by strain is completely different under simple shear loading. Also, the shear stress derived from the constitutive model based on strain invariants and principal stretchs has multiple relationships related to shear strain. The results in this study would be used to understand the more accurate mechanical characterization of soft tissue, which will allow us to evaluate the injury and develop much accurate injury criteria for soft tissue.

Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models

Abstract We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.

A model for multiphase moisture and heat transport below and above the saturation point of deformable and swelling wood fibers-II: Hygro-mechanical response

A non-linear elastic constitutive model for a swelling/shrinking orthotropic wood matrix is proposed. The model is thermodynamically consistent, derived based on the principles of continuum mechanics and the hybrid mixture theory. Moisture induced strains are introduced considering finite deformations by assuming a multiplicative split of the deformation gradient tensor into a swelling and an elastic part. Novel definitions of the Cauchy stress tensor and the moisture-dependent elastic material tangent matrix are obtained. The model is coupled to a multi-phase transient mass and heat transport model developed in Part I of this work. In this part of the work 2-D and 3-D test examples are used to describe the ability of the model to simulate moisture-induced distortions when drying wood within the hygroscopic and also from the over-hygroscopic moisture ranges. Despite deriving the model considering wood, the obtained constitutive relations can be suitably adopted to other orthotropic porous materials displaying properties similar to that of wood.

A beam theory for a new class of constitutive relation for elastic bodies

A theory for two-dimensional (2D) beams is presented for the case of considering new constitutive equations, wherein the linearized (infinitesimal) strain tensor is assumed to be a nonlinear function of the Cauchy stress tensor. An incremental formulation is developed to solve numerically the resultant equations. Three constitutive equations for elastic bodies are considered: a model wherein we have strain-limiting behavior, a nonlinear model for rock, and a bimodular constitutive equation for rock (which can also be considered as a nonlinear model for rock). Two boundary value problems are studied: the deformation of a cantilever beam with a point load, and a three-point beam with a point load applied on the middle. The numerical results are compared with the predictions of the classical theory for beams for the case of linearized elastic solids. The results for the stresses can be notoriously different, when comparing the predictions of the nonlinear constitutive equations with the results obtained using the linearized theory of elasticity.

Growth spin in plant cell wall growth and remodelling

A growth spin model is proposed for the modelling of the expansive growth and fibre remodelling of cell wall. The introduction of the growth spin allows to relax the perfectly bonding assumption in the kinematical growth, which provides a more sophisticated and flexible kinematical description for modelling the selective control and flexible regulation of anisotropic growth. Extended hardening laws are proposed for the growth and remodelling, respectively, aiming at further clarification of the dynamic balance between hardening and softening in a growing cell wall. The proposed model may shed a new light into the micro-structural interpretation of the “fictitious” intermediate (growth) configuration in the kinematical growth modelling of soft matter. A case study of the cell wall as a growing cylindrical wall is presented to demonstrate the proposed model. Spencer’s deviatoric stress tensor and its rate format are shown to play a key role in the modelling of the cell wall as a fibre-reinforced soft tissue.

Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate

Background: The infinite shear viscosity model of Carreau fluid characterizes the attitude of fluid flow at a very high/very low shear rate. This model has the capacity for interpretation of fluid at both extreme levels, and an inclined magnetic dipole in fluid mechanics has its valuable applications such as magnetic drug engineering, cold treatments to destroy tumors, drug targeting, bio preservation, cryosurgery, astrophysics, reaction kinetics, geophysics, machinery efficiency, sensors, material selection and cosmology. Novelty: This study investigates and interprets the infinite shear rate of Carreau nanofluid over the geometry of a cylindrical channel. The velocity is assumed to be investigated through imposing an inclined magnetic field onto cylindrical geometry. Activation energy is utilized because it helps with chemical reactions and mass transport. Furthermore, the effects of thermophoresis, the binary chemical process and the Brownian movement of nanoparticles are included in this attempt. Formulation: The mathematics of the assumed Carreau model is derived from Cauchy stress tensor, and partial differential equations (PDEs) are obtained. Similarity transformation variables converted these PDEs into a system of ordinary differential equations (ODEs). Passing this system under the bvp4c scheme, we reached at numerical results of this research attempt. Findings: Graphical debate and statistical analysis are launched on the basis of the obtained computed numerical results. The infinite shear rate aspect of Carreau nanofluid gives a lower velocity. The inclined magnetic dipole effect shows a lower velocity but high energy. A positive variation in activation energy amplifies the concentration field.

Spatial Constitutive Modeling of AA7050-T7451 with Anisotropic Stress Transformation.

The mechanical properties of anisotropic materials are generally characterized based on the orthotropy or transverse isotropy. However, the two-dimensional plane stress problems cannot comprehensively characterize the anisotropy of materials. In this study, based on the theory of elasticity and the transformation of the three-dimensional space coordinate system, combined with the projection relationship of the Cauchy stress tensor of an arbitrary section, the transformation relationship of the elastic modulus, shear modulus, and stress–strain between the orthogonal and load coordinate systems are obtained. The orthotropic Johnson-Cook (JC) constitutive model of AA7050-T7451 aluminum alloy is modified by fitting, and the constitutive relationship at any spatial angle is theoretically calculated by combining the obtained spatial coordinate transformation matrix. The generated spatial constitutive model is verified and modified through experiments. The results demonstrate that the theoretical mechanical properties and the modified spatial constitutive model can accurately reflect the effect of the spatial angle on the material stress distribution.

An exact consistent tangent stiffness matrix for a second-gradient model for porous plastic solids: Derivation and assessment

It is well known that the use of a consistent tangent stiffness matrix is critical to obtain quadratic convergence of the global Newton iterations in the finite-element simulations of problems involving elasto-plastic deformation of metals, especially for large-scale metallic structure problems. In this article, we derive an exact consistent stiffness matrix for a porous material model, the GLPD model developed by Gologanu, Leblond, Perrin, and Devaux for ductile fracture of porous solids based on generalized continuum mechanics assumptions. Full expressions for the derivatives of the Cauchy stress tensor and the generalized moments stress tensor the model involved are provided and implemented into a finite-element code. The effectiveness and robustness of the proposed tangent stiffness moduli are assessed by applying the formulation in the finite-element simulations of typical ductile fracture problems. Comparisons between the performance of our stiffness matrix and the standard ones are also provided.

Existence and uniqueness of solution of a thermoviscoelastic equation with time-dependent coefficients

In this paper, we prove the existence and uniqueness of the deformation of a semi-transparent body Ω resulting from its radiative heating by a black source S surrounding it at absolute temperature TS(t) for 0≤t≤tf. This deformation is modeled during the thermal process by Maxwell's model of viscoelasticity theory with a viscosity in the constitutive equation depending on the absolute temperature T(x,t) solution of the heat transfer equations between S and Ω. The rate of thermal deformations is also taken into account in the constitutive equation. We consider a mixed formulation of the thermoviscoelasticity problem in the variables (σ,v,ρ) where σ is the Cauchy stress tensor field, v the velocity field, and ρ the vorticity tensor field. For the proof of the well posedness of this mixed formulation, we will need to prove that ∂T∂t exists and is continuous on Q¯, where Q:=Ω×]0,tf[. This regularity result plays a critical role in proving the boundedness of a family of linear operators that leads directly to establishing the time regularity of the stress fields of a family of auxiliary elasticity problems. It is precisely these auxiliary stress fields that will allow us to find the unique stress field for the dual mixed formulation of our non-isothermal Maxwell model.

Comment on “Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli”

We show that elastic moduli of higher (third and fourth) order of prestressed single crystals at high pressure, obtained from the Gibbs free energy, enter the relation between the Cauchy stress and Lagrange finite-strain tensor the same way they do for the second-order elastic moduli and, therefore, directly describe the nonlinear elasticity of prestressed crystals.

Analysis of the propagation of small-amplitude waves in nonlinear elastic solids for problems with infinitesimal strains

A relatively new type of constitutive equation for elastic bodies, wherein the linearized strain tensor is a function of the Cauchy stress tensor, is studied for the particular case the stresses are divided into two parts. One is an initial time-independent stress to which is added an incremental (small in comparison with that initial stress) time-dependent stress. Incremental equations are derived that can be used to study the propagation of small-amplitude waves. The behaviour of such waves is studied for three boundary value problems, where the distributions of the initial stresses are inhomogeneous. The problems are: the inflation of a hollow sphere, the inflation of an infinitely long cylindrical annulus, and the circumferential shear and inflation of a cylindrical annulus. The behaviour of the small-amplitude waves is compared for different external loads, investigating how the initial stress affects such waves. A nonlinear model for a class of isotropic dry elastic rock is considered.