Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part III: Yield surface in tension–tension stress space (Al 6061–T 6511 and annealed 1100 Al)

Subsequent yield surfaces for aluminum alloys are determined for three proportional loading paths (i.e., axial, hoop, and combined hoop and axial stress) using 10 με deviation from linearity as the definition of yield. This paper is in continuation with Parts I and II of the author’s previous papers on subsequent yield surfaces under tension–torsion (σ11–√3σ₁₂) stress space using similar small offset definition of yield. In the current paper comprehensive experimental results on subsequent yield surfaces under tension–tension (σ₁₁–σ₂₂) stress space are presented. Comparison of subsequent yield surfaces under (σ₁₁–√3σ₁₂) stress space, investigated in the earlier papers, clearly indicated distinctive hardening behavior under various loading paths. However, subsequent yield surfaces for Al 6061–T 6511 (a low work hardening alloy) showed contraction and negative cross-effect with finite deformation as compared to the annealed 1100 Al (a high work hardening alloy) where expansion and positive cross-effect was observed.     

Determination of thermal conductivity and interfacial energy of solid Zn solution in the Zn-Al-Bi eutectic system

The equilibrated grain boundary groove shapes for solid Zn solution (Zn-3.0 at.% Al-0.3 at.% Bi) in equilibrium with the Zn-Al-Bi eutectic liquid (Zn-12.7 at.% Al-1.6 at.% Bi) have been observed from quenched sample with a radial heat flow apparatus. Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy for solid Zn solution in equilibrium with Al-Bi-Zn eutectic liquid have been determined to be (5.1 ± 0.4) × 10⁻⁸ K m, (80.1 ± 9.6) × 10⁻³ and (158.6 ± 20.6) × 10⁻³ J m⁻² from the observed grain boundary groove shapes, respectively. The thermal conductivity variation with temperature for solid Zn solution has been measure with radial heat flow apparatus and the value of thermal conductivity for solid Zn solution has been determined to be 135.68 W/km at the eutectic melting temperature. The thermal conductivity ratio of equilibrated eutectic liquid to solid Zn solution, R = K_L(Zn)/K_S(Zn) has also been measured to be 0.85 with Bridgman type solidification apparatus.     

Power-law creep and residual stresses in a carbopol gel

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ) ^α , with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ? 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.     

Yield criterion and elasto-plastic damage constitutive model for frozen sandy soil

A series of triaxial compression tests was carried out on a frozen sandy soil under confining pressures of 0–18 MPa at −6 °C. The experimental results indicate that, the strength of frozen sandy soil increases versus the increase in the confining pressures when σ₃ ? 3 MPa, but decreases when σ₃ > 3 MPa. This phenomenon is called the strengthening and weakening effects of confining pressures. A yield function, considering both effects, is proposed using the experimental method according to Drucker’s postulate, and the mathematical expression of the hardening parameter, which can describe the softening and hardening phenomenon, is provided. An elasto-plastic constitutive model for frozen sandy soil is developed. Based on the continuum damage theory, the cross anisotropic damage variables are deduced and their change regularities are investigated. Then the elasto-plastic damage constitutive model is proposed by introducing damage variables into elasto-plastic constitutive model. The validity of the model is verified by comparing its modeling results with experimental results obtained from triaxial tests. It is found that, this model can predict the deformation regularity of frozen soil exactly. It can simulate the stress–strain process under high confining pressures when pressure melting phenomena appear especially well.     

Deformation behavior and microstructure/texture evolution of an annealed 304 AISI stainless steel sheet. Experimental and micromechanical modeling

Different tensile tests are performed at −60 °C, on a 304 AISI stainless steel, to produce strain induced α′ martensite. Using EBSD analysis and X-ray diffraction methods, the corresponding texture and microstructure evolutions is studied, as well as the γ → α′ transformation mechanism. It is shown that α′ martensite is associated with {1 1 1}_γ faulted bands, partially ε bands, as often reported in the literature for low SFE stainless steels. As a consequence, these deformation bands play a key role in the nucleation and growth of α′ martensite. A micromechanical model is applied to simulate the deformation behavior of the stainless steel as well as the corresponding transformation kinetic and γ texture evolution. Different simulations are proposed, using various assumptions for the γ → α′ transformation mechanism. The simulation data are in better agreement with the experimental ones, when the γ → α′ transformation strain is derived from the Bowles–MacKenzie theory using the (1 1 1)_γ[−1 2 −1]_γ stacking fault shear system as lattice invariant shear.     

Multiaxial yield behaviour of Al replicated foam

Multiaxial experiments are performed on replicated aluminium foam using a custom-built apparatus. The foam structure is isotropic, and features open monomodal pores in average diameter. Plane stress (σ₁, σ₂, σ₃=0) and axisymmetric (σ₁, σ₂=σ₃) yield envelopes are measured using cubical specimens, supplemented by tests on hollow cylindrical and uniaxial samples. In addition to the three stress components at 0.2% offset strain, the computer-controlled testing apparatus also measures the three instantaneous displacement vectors.Results show that the shape of the yield surface is independent of the relative density of the foam in the explored range (13-28%). Strain increment vectors lie, within error, roughly normal to the line traced through data points in stress space. Replicated foams feature asymmetric yield behaviour between tension and compression. The data additionally show an influence on the yield surface of the third stress tensor invariant (i.e., of the Lode angle). Simple general expressions for the yield surface are fitted to the data, leading to conclude that their behaviour is slightly better captured by parabolic rather than elliptic expressions dependent on all three stress invariants.     

Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics

Triaxial compressive tests of frozen silt were carried out under confining pressures from 0.0 to 14.0 MPa at the temperatures of −2, −4 and −6 °C. A strength criterion based upon experimental results is presented by the combination of extended Lade–Duncan strength function f_π(θ) in π-plane and f_p–q(p) in p–q-plane. In order to describe the deformation characteristic of frozen silt, an elastoplastic constitutive model in generalized plastic mechanics has been proposed for the nonlinear behavior of frozen silt, such as the pressure melting and crushing phenomena, strain softening/hardening characteristics and dilatation, etc., by employing an elliptical yield surface, together with a non-associated flow rule for the compressive mechanism, and two parabolic yield surfaces, together with non-associated flow rules for the shear mechanism. The validity of the model is verified by comparing its modeling results with the results of triaxial compressive tests. It is found that the stress–strain curves predicted by this model agree well with the corresponding experimental results both under low and high confining pressures.     

Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

A note on the Mohr-Coulomb and Drucker-Prager strength criteria

This paper presents different expressions of the Mohr-Coulomb (M-C) criterion as well as the interrelationships between them, which lays a foundation for the definition of the equivalent M-C friction angle φ_mc. The characteristics of four types of Drucker-Prager cones matched with the M-C surface are compared as the friction angle φ varies from 0° to 90°. The minimum and maximum value of φ for them is given and the influence of the intermediate principal stress σ₂ to the major principal stress σ₁ is demonstrated using their φ_mc.     

Dynamic effective shear strength of saturated sand

The dynamic effective shear strength of saturated sand under cyclic loading is discussed in this paper. The discussion includes the transient time dependency behaviors based on the analysis of the results obtained in conventional cyclic triaxial tests and cyclic torsional shear triaxial tests. It has been found that the dynamic effective shear strength is composed of effective frictional resistance and viscous resistance, which are characterized by the strain rate dependent feature of strength magnitude, the coupling of consolidation stress with cyclic stress and the dependency of time needed to make the soil strength sufficiently mobilized, and can also be expressed by the extended Mohr-Coulomb's law. The two strength parameters of the dynamic effective internal frictional angle φ_d and the dynamic viscosity coefficient η are determined. The former is unvaried for different number of cyclic loading, dynamic stress form and consolidation stress ratio. And the later is unvaried for the different dynamic shear strain rate developed during the sand liquefaction, but increases with the increase of initial density of sand. The generalization of dynamic effective stress strength criterion in the 3-dimensional effective stress space is studied in detail for the purpose of its practical use. The project supported by the National Natural Science Foundation of China (10172070)     

A new unified constitutive model with short- and long-range back stress for lead-free solders of Sn–3Ag–0.5Cu and Sn–0.7Cu

A series of tensile tests of Sn–3Ag–0.5Cu and Sn–0.7Cu lead-free solders were investigated at various strain rates from 1 × 10⁻⁴ s⁻¹ to 1 × 10⁻² s⁻¹ and over a wide temperature range from 25 ^oC to 150 ^oC. Two-step strain rate jump tests, three-step short term creep tests with stress jump, and uniaxial ratcheting tests were also conducted. Based on the test data, a new constitutive model was proposed with a simple formulation and only eight material constants which can be easily obtained. The model employs two carefully defined back stress components to simulate the loading/unloading asymmetry phenomenon in uniaxial ratcheting tests. Different evolution rules of short-range back stress were given for loading and unloading stage, which provides the model ability to simulate the asymmetry in hysteresis loops. The proposed model presents good simulation of uniaxial tensile tests, strain rate jump tests, short term creep tests with stress jump, and uniaxial ratcheting tests.     

Screw dislocations and an interfacial blunt crack with viscoelastic interface

This work deals with the influence of Kelvin-type viscoelastic interface on the generation of screw dislocations near the interfacial blunt crack tip in light of a pair of concentrated loads. The stress fields for dislocation and concentrated load have been obtained by using the integral transform and conformal mapping, the stress intensity factor have been studied, the image force acting on dislocation has been analyzed. The region r_b where n screw dislocations are generated by a pair of concentrated loads and dislocation number are obtained by displacement compatibility and stress compatibility conditions of self-consistent and self-equilibrated systems. The results show that: the force acting on dislocation starts with the value that a perfectly bonded interface, then with relaxation of the imperfect interface; the shield effect for dislocation decreases as time goes by; in addition, with time elapsing, the influence of material shear modulus rate on shielding effect becomes weaker and weaker. The scale of multiplier α(r_b/a) increases with relaxation of imperfect interface, the larger ratio of crack geometry c/a and the smaller ratio of shear modulus μ₁/μ₂ will lead the higher scale of multiplier. When μ₁/μ₂ = 1, the screw dislocations number first increases and then decreases with relaxation of imperfect interface, In addition, it possesses the highest value at t₀ ≈ 1 and tends to vanish at t₀ = ∞. When μ₁ < μ₂, the screw dislocations number increases with relaxation of imperfect interface. When μ₁ > μ₂, the screw dislocations number first increases then decreases with relaxation of imperfect interface, and possesses the highest value at t₀ ≈ 1, the negative value are exclude from the discussion.     

An experimental investigation on isoclinic parameters in viscoelastoplastic materials under cyclic stresses

Photovisco-elastoplasticity has been developed and applied to various problems in nonlinear stress analysis as an extension of photoelasticity into nonlinear photomechanics. In photoelasticity, because stress is simply related to strain, isochromatics are proportional to either the principal stress difference or the principal strain difference and isoclinics indicate the directions of either principal stresses or principal strains separately. However, in photovisco-elastoplasticity, these optical responses are nonlinearly related to both stress and strain state simultaneously. Thus isochromatics are related nonlinearly with both the principal stress and strain differences. Isoclinics depend on the direction of principal strain as well as that of principal stress. Concerning now isoclinic parameter: according to the results of experiments performed by us for the deformation state where the directions of principal stress and strain do not coincide with one another, it has been found that isoclinic parameter moves gradually from the direction of principal stress to that of principal strain with increasing viscous deformation under constant stress. In the present research, extending our previous investigations, the behavior of isoclinic parameter has been examined experimentally under cyclic stressing, where the direction of principal stress changes alternately. In the course of an experiment, the variation of isoclinic parameter in relation to the number of cycles was measured together with the corresponding strain state on the thin-walled tubular specimen of celluloid softened by heating, subjected to combined loading conditions consisting of constant axial tension and cyclic torsion. The results obtained may be summarized as follows.

1. The cyclic-variation phase of isoclinic parameter lags behind the cyclic-variation phase of the direction of principal stress. The phase difference between these cyclic variations remains constant regardless of the number of cycles.
2. Isoclinic parameter has a value close to the direction of the principal stress within a range of a small number of cycles and decreases with an increase in the number of cycles. However, it does not approach the direction of principal strain but a value between the direction of the principal stress and that of the principal strain, with a certain constant ratio to them.

     

A stress analytical solution for Mode III crack within modified gradient elasticity

The strain gradient exists near a crack tip may significantly influence the near-tip stress field. In this paper, the strain gradient and the internal length scales are introduced into the basic equations of mode III crack by the modified gradient elasticity (MGE). By using a complex function approach, the analytical solution of stress fields for mode III crack problem is derived within MGE. When the internal length scales vanish, the stress fields can be simplified to the stress fields of classical linear elastic fracture mechanics. The results show that the singularity of the shear stress is made up of two parts, r^−1/2 part and r^−3/2 part, and the sign of the stress σ_yz changes. With the increase of l_x, the peak value of σ_yz decrease and its location moves farther from the fracture vertex. The influence of strain gradient for mode III crack problem cannot be ignored.     

Thermo-mechanical response of Al 6061 with and without equal channel angular pressing (ECAP)

The thermo-mechanical responses of Al 6061 before and after equal channel angular pressing (ECAP) at different strain rates and temperatures were measured. Al 6061 was solution heat treated before ECAP pressing at room temperature and subjected to up to three passes. After pressing, the billets were aged at 100 °C for 2 days. An as-received Al 6061-T651 was studied similarly to investigate the differences between processed and non-processed specimens. The responses of ECAP material were determined at −30, 22, 125 and 250 °C, and at strain rates from 10⁻⁵ to 2530 s⁻¹; the 6061-T651 specimens were subjected to uniaxial compressive loading at −31, 22, 85, 150, 230 and 315 °C, and strain rates ranging from 10⁻⁵ to 2200 s⁻¹. It was found that, the ECAP process increases the strength versus the T651 condition. Additionally, the Al 6061 ECAP is not sensitive to strain rate at room and lower temperatures, but the sensitivity increases as the number of passes and/or temperature are increased and this is the same for the non-processed material. Increasing the number of passes increases the flow stress at room and lower temperatures, has almost no effect at 125 °C and decreases at 250 °C. For both materials, the dynamic flow stress is higher than the stress at quasi-static strain rates even when the quasi-static strain rate regime is insensitive to strain rate. The Al 6061 has strong texture after one pass but steadily increases as the number of passes are increased. This is the first study that reports on the thermo-mechanical responses of ECAP and non-ECAP Al 6061 at such a wide range of strain rates, including dynamic, and temperatures.     

Tensile-shear transition in mixed mode I/III fracture

The propensity of the transition of fracture type in either brittle or ductile cracked solid under mixed-mode I and III loading conditions is investigated. A fracture criterion based on the competition of the maximum normal stress and maximum shear stress is utilized. The prediction of the fracture type is determined by comparing τ_max/σ_max at a critical distance from the crack tip to the material strength ratio τ_C/σ_C, i.e., (τ_max/σ_max)<(τ_C/σ_C) for tensile fracture and (τ_max/σ_max)>(τ_C/σ_C) for shear fracture, where σ_C (τ_C) is the fracture strength of materials in tension (shear). Mixed mode I/III fracture tests were performed using circumferentially notched cylindrical bars made of PMMA and 7050 aluminum alloy. Fracture surface morphology of the specimens reveals that: (1) for the brittle material, PMMA, only tensile type of fracture occurs, and (2) for the ductile material, 7050 aluminum alloy, either tensile or shear type of fracture occurs depending on the mode mixity. The transition (in ductile material) or non-transition (in brittle material) of the fracture type and the fracture path observed in experiments were properly predicted by the theory. Additional test data from open literature are also included to validate the proposed theory.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

A STRESS VECTOR-BASED CONSTITUTIVE MODEL FOR COHESIONLESS SOIL (Ⅱ)-APPLICATION

The stress vector-based constitutive model for cohesionless soil, proposed by SHI Hong-yan et al., was applied to analyze the deformation behaviors of materials subjected to various stress paths. The result of analysis shows that the constitutive model can capture well the main deformation behavior of cohesionless soil, such as stress-strain nonlinearity, hardening property, dilatancy, stress path dependency, non-coaxiality between the principal stress and the principal strain increment directions, and the coupling of mean effective and deviatoric stress with deformation. In addition, the model can also take into account the rotation of principal stress axes and the influence of intermediate principal stress on deformation and strength of soil simultaneously. The excellent agreement between the predicted and measured behavior indicates the comprehensive applicability of the model. Biography: SHI Hong-yan (1962-), Associate Professor, Doctor     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.