A study on rate-type constitutive equations and the existence of a free energy function

Summary Demanding that rate-type constitutive equations in solid mechanics has to be in accord with the thermodynamical requirement of the existence of a free energy function, the restrictions to be imposed on stress evolution equations in terms of theLie (Oldroyd) as well as theZaremba-Jaumann objective rate of theCauchy stress tensor are discussed. Explicit forms of generalized constitutive equations are given according to which the tangent modulus relating the objective rate of the stress tensor to the strain velocity tensor is no more constant but assumed to depend on the stress state itself. The use of theKirchhoff stress tensor is handled as well. It is shown that theZaremba-Jaumann derivative necessarily reduces to theLie one. As an application the case of simple shear is discussed where a monotonously increasing stress-strain relation is obtained. The paper closes with remarks concerning elasto-plasticity.     

Stress state and the static strength of mechanically inhomogenous welded joints. Report 2. Effect of a small degree of mechanical inhomogeneity on the strength of butt welded joints

The characteristics of the stress state in the presence of a small amount of mechanical inhomogeneity (k_u<2) have been uncovered for welded joints containing a soft interlayer. An approach has been developed for estimating the weight carrying ability of these joints. The association of the contact tangential stresses with the basic structural-geometric parameters of a welded joint (k_u, k) under conditions of plane strain has been established based on the mathematical theory of plasticity. More accurate formulas describing the stress tensor components and the mean stresses during ductile failure have been presented. These formula are in good agreement with the experimental data. In assuming an equilibrium distribution of the strength characteristics over the volume of the soft interlayer, expressions were obtained which can be used at the design stage for choosing the optimum parameters for the welded joint, thus increasing its reliability.Translated from Problemy Prochnosti, No. 4, pp. 35–38, April, 1991.     

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C _d≡ F/(ηU) = k ₀ + k ₂ Re²+ .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K (U/a)^n-1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k ₀ is the drag coefficient for creeping flows. The proportionality constant k ₂ depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k ₂ on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.     

Study of the effects of stress and strain on martensite transformation: Kinetics and transformation plasticity*

In this paper, the effects of stress and strain on the kinetics and plasticity during martensitic transformation are studied. The mathematical models of transformation kinetics and plasticity under stress are developed. According to experimental results, the transformation plasticity parameter k is concluded not to be a constant, but it varies with the stresses.     

Experimental results on the influence of thermal stresses on the stability of a crack

The case of the stability of a crack under uniform tension and thermal stresses is studied. The thermal stresses are caused by the uniform heat flow disturbed by the insulated crack. The elastic stress distribution in the vicinity of a crack tip and the stress intensity factors k ₁, k ₂ are evaluated. The Barenblatt fracture criterion is applied, and the function f(k ₂/k ₁) is found which determines the variation of the critical tension stresses caused by the thermal stresses. Experiments were carried through that have verified the form of the function f(k ₂/k ₁).

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Zusammenfassung Die Stabilität eines Risses, welcher gleichmassigen mechanicher- und Wärmespannungen unterworfen ist, wird untersucht. Wärmespannungen werden durch Störung des gleichmässigen Wärmestromes durch den isolierten Riß, hervorgerufen. Die elastische Spannungsverteilung in der näheren Umgebung des Risses und die Spannungsintensitätsfaktoren k ₁, k ₂ werden geschätzt. Es wird der Barenblatt Bruchkennwert angewendet und die Funktion f(k ₂/k ₁), welche die Änderung der durch Wärmespannungen hervorgerufenen kritischen Spannungsbeanspruchung bestimmt, wird ermittelt.Die gültigkeit der Form der Funktion f(k ₂/k ₁) wurde durch Versuche bestätigt.

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Résumé On étudie le cas d'une fissure en état stable sous un champ de contraintes thermiques uniformes. Ces contraintes thermiques sont dues a un flux de chaleur uniforme auquel une fissure isolée confère une perturbation. On évalue la distribution des contraintes élastiques au voisinage de l'éxtrémité de la fissure, ainsi que les facteurs d'intensité de contrainte k ₁ et k ₂.Le critère de rupture de Barenblatt est appliqué, et l'on détermine la fonction f(k ₂/k ₁) qui gouverne la variation des tensions critiques produites par les contraintes thermiques.La forme de la fonction f(k ₂/k ₁) est vérifiée par voie expérimentale.

     

Determination of the characteristic directions of lossless linear optical elements

We show that the problem of finding the primary and secondary characteristic directions of a linear lossless optical element can be formulated in terms of an eigenvalue problem related to the unimodular factor of the transfer matrix of the optical device. This formulation makes any actual computation of the characteristic directions amenable to pre-implemented numerical routines, thereby facilitating the decomposition of the transfer matrix into equivalent linear retarders and rotators according to the related Poincaré equivalence theorem. We explain in detail how this issue arises in the context of stress analysis based on integrated photoelasticity or hybrid methods combining photoelastic measurements with analytical stress models and/or numerical Finite-Element computations for the stress tensor field. Furthermore we show how our results can be applied when algorithms for the reconstruction of the dielectric tensor in the interior of a photoelastic model (dielectric tensor imaging) are tested for their stability against noise in the measurement data. For the sake of completeness we provide a brief derivation of the basic equations governing integrated photoelasticity.     

Tensor models of rupture strengh. Report no. 1. Steady loading of initially isotropic and anisotropic bodies

A relationship is established between the rupture-strength theory of Il'yushin and the Pisarenko—Lebedev criterion. The variant proposed for constructing tensor models of rupture strength is based on prior knowledge of the rupture-strength surface — the dependence of the time to fracture under steady loading on stress intensity and other dimensionless invariants of the stress state. This condition makes it possible to reduce the damage tensor to a deviator, so that simpler criterional relations follow from the model. The rupture-strength surfaces of certain steels and alloys are constructed from well-known experimental data obtained for a plane stress state. Tensor models for bodies with initial rupture-strength anisotropy are also constructed. Satisfactory agreement is obtained between theoretical and experimental data on the tension of single crystals of a nickel alloy with an fcc lattice.Translated from Problem Prochnosti, No. 8, pp. 76–80, August, 1995.     

Salt effect of lithium bromide on the solution polymerization of acrylonitrile

Abstract

In this paper, the salt effect of lithium bromide (LiBr) on the dimethyl formamid (DMF) solution polymerization of acrylonitrile was studied. From the experimental data, we proposed a possible mechanism and derived its rate expression to determine the kinetic parameters for this system. The results show that the values of k_v/k_tr (1.984～79.365) and k_t/k² _tr (3.311×l0²～3.156×l0⁵ sec·mole/1) of the DMF‐LiBr solution polymerization of acrylonitrile are larger than those of the DMF‐LiCl solution polymerization of acrylonitrile shown by Bamford and others. These results can be explained by the basicity of salt and the dissociation of polyacrylonitrile in the DMF‐LiBr solution.     

The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain problem

The effect of a homogeneous cylindrical inlay on cracks in the doubly-periodic complete plane strain (CPS) problem is investigated in this paper. By employing the solutions of doubly quasi-periodic and doubly-periodic Riemann boundary value problems for analytic functions we obtain the general solution in closed form. And approximate analytical expression of the stress intensity factors, which are identical with the known results, are obtained when there are no holes or gaskets in the periodic parallelogram and e ₃=0, F _k=0, T _k=0, k=1,2, with constant load applied at the edges of the crack.     

An analytical expression for the J2‐integral of an interfacial crack in orthotropic bimaterials

This paper presents a new analytical expression relating the J₂‐integral and stress intensity factors (SIF) in an in‐plane traction‐free crack between two orthotropic elastic solids using the complex function method. The singular oscillatory near tip field of a bimaterial interfacial crack is usually characterized by a pair of SIFs. In linear elastic interfacial fracture mechanics, the majority of numerical and experimental methods rely on the analytical equations relating J_k‐integrals and SIFs. Although an analytical equation relating J₁‐integral or strain energy release rate and SIFs is available, a similar relation for J₂‐integral in debonded anisotropic solids is non‐existent. Using this new analytical expression, in conjunction with the values of J_k, the SIFs can be computed without the need for an auxiliary relation. An example with known analytical solutions for SIFs is presented to show the variation of the J₂‐integral near the crack tip of a bimaterial orthotropic plate. Different bimaterial combinations are considered, and the effect of material mismatch on J_k is demonstrated.     

The Prediction of the Dissolution Rate Constant by Mixing Rules: The Study of Acetaminophen Batches

The purpose of this article is to promote two simple and scalable methods to accelerate the formulation development of formulated granules using acetaminophen as a model system. In method I, formulated granules made from the batch of small particle–sized acetaminophen (1) by ball milling the batch of large particle–sized acetaminophen (2), and the mixture of the two batches at equal weights (mix) gave the dissolution rate constants (k) of k₁?=?0.43 ± 0.15 minutes^?1, k₂?=?0.18 ± 0.01 minutes^?1, and k_mix?=?0.30 ± 0.03 minutes^?1 for 75 wt percent formulation; k₁?=?0.75 ± 0.01 minutes^?1, k₂?=?0.18 ± 0.01 minutes^?1, and k_mix?=?0.34 ± 0.03 minutes^?1 for 62 wt percent formulation; and k₁?=?0.28 ± 0.01 minutes^?1, k₂?=?0.16 ± 0.01 minutes^?1, and k_mix?=?0.22 ± 0.02 minutes^?1 for 30 wt percent formulation. In method II, the mixture of the formulated granules produced by mixing the formulated granules from the two batches at equal weights gave dissolution rate constants of k_mix?=?0.30 ± 0.03 minutes^?1, 0.30 ± 0.02 minutes^?1, and 0.22 ± 0.01 minutes^?1 for 75 wt percent, 62 wt percent, and 30 wt percent formulations, respectively. After fitting the three data points of k₁, k₂, and k_mix to the 10 mixing rules in materials science—series mixing rule, Hashin and Shtrikman upper bound, logarithmic mixing, Looyenga mixing rule, effective media approximation (EMA), three-point lower bound, Torquato approximation, three-point upper bound, Maxwell mixing rule, and parallel mixing rule—we found that the selection of the best suited mixing rules based on k₁, k₂, and k_mix was solely dependent on the formulations under a given operating condition and regardless of whether the system was a powder mixture or a granular mixture. The values of k₁, k₂, and k_mix in both the 75 wt percent and 30 wt percent formulations were enveloped by the parallel mixing rule and Maxwell mixing rule, whereas the values of k₁, k₂, and k_mix for the 62 wt percent formulation were encompassed by the logarithmic mixing rule, Hashin and Shtrikman upper bound, and the series mixing rule. Apparently, the best suited mixing rules could be used to predict the right proportions of either the powder mixture (Method I) or the granular mixture (Method II) for obtaining any other desired dissolution rate constant, k_mix, whose value fell in between the values of k₁ and k₂.     

The Nature of Stress and Electric-displacement Concentrations around a Strongly Oblate Cavity in a Transversely Isotropic Piezoelectric Material

In this paper we explore the nature of stress and electric-displacement concentrations around a strongly oblate spheroidal cavity that possesses a finite dielectric permittivity. We start out from Eshelby’s general inclusion method but give specific account on the important class of piezoelectric ceramics whose structure is represented by the 6 mm symmetry. It is found that under axial electromechanical loading these concentrations are governed by a dimensionless parameter η, defined as (k₀/k₃₃)/(c/a), that involves the ratio of the dielectric permittivity of the medium inside the cavity k₀, to that of the transversely isotropic piezoelectric ceramic k₃₃, and the aspect ratio of the cavity c/a. When the medium inside the cavity is an impermeable one it is found that both the axial stress and axial electric displacement can have direct contribution to the concentration factors, but when the medium is a conducting one only the applied stress has an effect on it. Our analysis further indicates that it is the parameter η – not k₀/k₃₃ or c/a alone – that plays the key role here; when η< 0.01, the cavity can be effectively treated as an impermeable one, while for η> 100 it can be treated as a conducting case. Numerical results for several PZT ceramics suggest that under a pure tensile stress the ceramic tends to fracture on the equatorial plane, but under a pure electrostatic load it tends to develop radial cracks normal to the edge of the cavity.     

On stresses conjugate to Eulerian strains

Summary. A stress is considered conjugate to a strain if the product of the stress and an objective rate of the strain has a trace which is equal to the rate of work per unit volume. Using Kronecker product relations, apparently new expressions for stresses conjugate to the Finger strain B, the Euler strain , the Eulerian (right) stretch tensor V, and log(V) are determined. In addition, a nonclassical strain p is introduced which permits a constitutive equation expressing its Truesdell rate in terms of B and the Truesdell rate of the Cauchy stress. We regard a tensor as a strain if (a) it is not affected by rigid body motion and (b) its current value, given suitable compatibility conditions, determines the current displacement field to within a rigid body translation and rotation. By these criteria e is not strictly a strain and instead we later refer to it as a pseudostrain.     

Dual-mixed hp finite element methods using first-order stress functions and rotations

 A two-field dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satisfied a priori by introducing the tensor of first-order stress functions. It is pointed out that the use of six properly chosen first-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacement-pressure formulation of elasticity and the velocity-pressure formulation of Stokes flow. Selection of stable mixed hp finite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p finite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp finite elements is presented and compared to displacement-based hp finite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h- and p-extensions. Received 22 October 1999     

Bending of a Layer with Through Tunnel Cuts and Free-Sliding End Faces

We propose a new procedure for the solution of mixed three-dimensional skew-symmetric problems of the theory of elasticity for a layer weakened by through tunnel cracks. The boundary-value problem is reduced to a system of 3k (k = 1, 2, ...) one-dimensional singular integral equations. We present some results of numerical calculations of the stress intensity factors.     

Friction tensor concept for textured surfaces

Directionality of grinding marks influences the coefficient of friction during sliding. Depending on the sliding direction the coefficient of friction varies between maximum and minimum for textured surfaces. For random surfaces without any texture the friction coefficient becomes independent of the sliding direction. This paper proposes the concept of a friction tensor analogous to the heat conduction tensor in anisotropic media. This implies that there exists two principal friction coefficients μ_1,2 analogous to the principal conductivities k _1,2. For symmetrically textured surfaces the principal directions are orthogonal with atleast one plane of symmetry. However, in the case of polished single crystalline solids in relative sliding motion, crystallographic texture controls the friction tensor.     

The null distribution of the likelihood-ratio test for one or two outliers in a normal sample

For testingk upper or lower outliers in a normal sample, the sampling distribution of the likelihood-ratio statistic is still unknown in the literature except fork=1. In this paper, we find its exact distribution fork=2 and tabulate the extensive critical values, which are compared with the traditional simulated values. The research is partially supported by the National Sciences and Engineering Research Council of Canada.     

Tables of the Distribution of the Mann-Whitney-Wilcoxon U-Statistic Under Lehmann Alternatives

A simple recursive formula is used to generate the distribution of the MWW U-statistic under alternatives of the form K ₁: G – F ^k ,1 ≠ k < 0, and K ₂: G – 1? (1 ? F) ^k , 1 ≠ k > 0, and it is pointed out that many of these alternatives are of interest in life testing, reliability testing and related fields. Tables are given for the case G = 1 – (1 ? F) ^k for 4 ≤ n, m ≤ 8 and k = 2,3,4,6,9,12 and it is shown that these tables can also be used when G = F ^k and when 0 < k < 1.     

On Minimum-Point Second-Order Designs

In this note, we discuss k-factor, second order designs with minimum number of points ½(k + l)(k + 2), in particular, those which are extensions of designs that give minimum generalized variance for k = 2 and 3. The experimental region is the unit cuboid. Minimum point designs of this type are unknown for k ≥ 4, and these designs are the best found to date except for k = 4, where a better design is known. Kiefer has shown that these designs cannot be the best for k ≥ 7, via an existence result but, even here, specific better designs are not known and appear difficult to obtain. We also discuss some difficulties of using, in practice, designs that, are D-optimal (that is give minimum generalized variance when the number of points is not restricted).