Large strain deformation of polycrystalline metals at low homologous temperatures

Large strain compression data (true strains to about ?3.0) are presented for polycrystalline α U and α Fe at room temperature. The results, together with other published data at low homologous temperatures (≈0.2 T_m), where T_m is the absolute melting temperature, suggest that a steady-state flow stress σ_s is approached after extensive strain-hardening, α U exhibits a very high strain-hardening rate, with σ_s ≈ 2900 MPa (420 ksi) indicating that cold-working is a very potent method of strengthening this metal. All the data evaluated can be fit by the stress-strain relation σ = σ_s? exp (?(Nε)^p)(σ_s? σ_y), where σ_y is the yield stess, p is a constant equal to a for the metals analyzed, N is a constant associated with the strain-hardening characteristics of a material, σ is true stress, and ε is true strain.     

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.     

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.     

Crack-tip fields for fast fracture of an elastic-plastic material

An asymptotic analysis of the near-tip fields is given for transient crack propagation in an elastic-plastic material. The material is characterized by J₂ flow theory together with a bilinear effective stress-strain curve. Both plane stress and plane strain conditions have been considered. Explicit results are given for the order of the crack-tip singularity, the angular position at which unloading occurs, and the angular variation of the near-tip stresses, all as functions of the crack-tip speed and the ratio of the slopes of the two portions of the bilinear stress-strain relation. It was found that the results are much more sensitive to the elastic-plastic constitutive relation than to the crack speed. This result is important for numerical analyses of dynamic crack propagation problems.     

Non-coaxiality of strain increment and stress directions in cross-anisotropic sand

An experimental program was carried out in a recently developed torsion shear apparatus to study the non-coaxiality of strain increment and stress directions in cross-anisotropic deposits of Fine Nevada sand. Forty-four drained torsion shear tests were performed at constant mean confining stress, σ_m, constant intermediate principal stress ratios, as indicated by b = (σ₂ − σ₃)/(σ₁ − σ₃), and constant principal stress directions, α. The experiments were performed on large hollow cylinder specimens deposited by dry pluviation and tested in an automated torsion shear apparatus. The specimens had height of 40 cm, and average diameter of 20 cm, and wall thickness of 2 cm. The stress–strain behavior of Fine Nevada sand is presented for discrete combinations of constant principal stress direction, α, and intermediate principal stress. The effects of these two variables on the non-coaxiality are presented. The experiments show that the directions of the strain increments do not in general coincide with the directions of stresses, and there is a switch from one to the other side between the two quantities.     

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.     

On the relationship between critical tensile stress and fracture toughness in mild steel

An analysis is presented which relates the critical value of tensile stress (σ_f) for unstable cleavage fracture to the fracture toughness (K_Ic) for a high-nitrogen mild steel under plane strain conditions. The correlation is based on (i) the model for cleavage cracking developed by E. Smith and (ii) accurate plastic-elastic solutions for the stress distributions ahead of a sharp crack derived by J. R. Rice and co-workers. Unstable fracture is found to be consistent with the attainment of a stress intensification close to the tip such that the maximum principal stress σ_yy exceeds σ_f over a characteristic distance, determined as twice the grain size. The model is seen to predict the experimentally determined variation of K_Ic with temperature over the range -150 to -75°C from a knowledge of the yield stress and hardening properties. It is further shown that the onset of fibrous fracture ahead of the tip can be deduced from the position of the maximum achievable stress intensification. The relationship between the model for fracture ahead of a sharp crack, and that ahead of a rounded notch, is discussed in detail.     

Crack-tip fields in steady crack-growth with linear strain-hardening

Singular stress and strain fields are found at the tip of a crack growing steadily and quasi-statically into an elastic-plastic strain-hardening material. The material is characterized byJ₂ flow theory together with a bilinear effective stress-strain curve. The cases of anti-plane shear, plane stress and plane strain are each considered. Numerical results are given for the order of the singularity, details of the stress and strain-rate fields, and the near-tip regions of plastic loading and elastic unloading.     

Wall slip and multi-tier yielding in capillary suspensions

Impact of wall slip on the yield stress measurement is examined for capillary suspensions consisting of cocoa powder as the dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid using smooth and serrated parallel plates. Using dynamic oscillatory measurements, we investigated the yielding behavior of this ternary solid-fluid-fluid system with varying particle volume fraction, ?, from 0.45 to 0.65 and varying water volume fraction, ?_w, from 0.02 to 0.08. Yield stress is defined as the maximum in the elastic stress (G^′γ), which is obtained by plotting the product of elastic modulus (G^′) and strain amplitude (γ) as a function of applied strain amplitude. With serrated plates, which offer minimal slippage, capillary suspensions with ? ≥?0.45 and a fixed ?_w =?0.06 showed a two-step yielding behavior as indicated by two peaks in the plots of elastic stress as a function of strain amplitude. On the other hand with smooth plates, the capillary suspensions showed strong evidence of wall slip as evident by the presence of three distinct peaks and lowered first yield stresses for all ? and ?_w. These results can be interpreted based on the fact that a particle-depleted layer, which is known to be responsible for slip, is present in the vicinity of the smooth surfaces. The slip layer presents itself as an additional “pseudo-microstructure” (characteristic length scale) besides the two microstructures, aqueous bridges and solid particle agglomerates, that may occur in the system. With serrated plates, both the yield stresses (σ₁σ₂) and storage moduli plateau at lower strain (before the first yield point) and at higher strain (before the second yield point) (G\(^{\prime }_{p1}\), G\(^{\prime }_{p2}\)) were found to increase with ? (at a fixed ?_w =?0.06) following power-law dependences. Similarly with increasing ?_w (0.02 – 0.08) at a fixed ? =?0.62, the system behaved as a solid-like material in a jammed state with particles strongly held together as manifested by rapidly increasing σ₁ and σ₂. The usage of smooth surfaces primarily affected σ₁ which was reflected by an approximately 70–90% decrement in the measured σ₁ for all values of ?. By contrast, σ₂ and G\(^{\prime }_{p2}\) were found to be unaffected as shown by close agreement of values obtained using serrated geometry due to vanishing slip layers at higher strain amplitudes.     

Influence of transverse normal stress on sheet metal formability

Since the late 1960's, the strain space forming limit diagram, FLD_ϵ, has been nearly universally employed as a method to help predict sheet metal failure for the open die stamping process. Traditional FLD_ϵ's in use today are based upon a plane stress assumption. However, a plane stress assumption may not always be valid for many sheet metal forming operations. Through the employment of a strain-to-stress space mapping procedure, a new sheet metal formability model that takes into account the through-thickness normal stress (σ₃) is proposed. Good agreement with a limited set of experimental data is found.     

Effect of sample dimensions,lubrication and deformation rate on uniaxial compression of gelatin gels

The response of 10% gelatin gels to uniaxial compression is determined in part by frictional effects at the gel-platen interface. By using teflon-coated plates, lubricated with paraffin or silicone oil, these frictional effects are effectively eliminated. The stress-strain response can then be described by the two-constant Mooney-Rivlin relation, the sum of the two parameters (C ₁ +C ₂ ) being about 25% lower in lubricated compression than the value obtained in simple shear and torsion. Cross-head speed (for total testing times of 0.2–3 min) had no effect on material response, but long-term stress relaxation does occur over periods of about 30 min and longer. Sample radius did not affect the response in lubricated compression but had a major effect under unlubricated conditions. No systematic change in response was seen with sample diameter to height (aspect) ratios between 9.6 and 3.1 in lubricated compression, but data scatter for a given sample diameter was worst at the lowest heights (highest aspect ratio). Agreement of all true stress versus strain data was within about ± 7% regardless of sample height or deformation rate.     

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.     

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.     

Probabilistic design and quality control in probabilistic strength of materials

A numerical simulation method is used here for the design and quality control of a material subject to normal gradual stress σ or a cyclic stress σ, having fixed cumulative probability F and the number of cycles l; capable of achieving a given mechanical property such as yield point, elastic limit stress, fracture strength, etc., as well as the admissible tolerance δF the presence of such property is to be accepted with. With F and δF, the stress of the design σ_C can be determined, as well as the variations δm, δσ₀ and δσ_L of Weibull’s parameters m, σ₀ and σ_L, respectively, that the tolerance δF admits. When cyclic stresses arise, other parameters must be introduced, k₁, k₂ and p, which produce variations δk₁, δk₂, and δp, respectively. The determination of the necessary number of samples to be tested in order to carry out the quality control of the material, with a given probability of effectiveness, is obtained with variations δm, δσ₀, δσ_L, δk₁, δk₂ and δp, with the parameter dispersion estimated by numerical simulation, and with the help of a property deduced from the Fischer’s matrix to obtain the parameter dispersion.     

Analytical solutions for finite cylinders compressed between platens and the strain effect on the valence-band structure of Si1−xGex alloy

This paper presents an analytical solution for inhomogeneous strain and stress distributions within finite circular cylinders of Si_1−xGe_x alloy under compression test with end friction. The method follows Lekhnitskii’s stress function approach, but a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. Numerical results show that the axial, radial, circumferential and shear strains are all inhomogeneous within finite cylinders, and local strain concentrations near two end surfaces were usually developed as long as friction exists between end surfaces and loading platens. Moreover, by using envelope-function method, the effect of strain on the valence-band structure of Si_1−xGe_x alloy is also studied. It was found that strain can induce band splitting, alteration of the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1−xGe_x alloy.     

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.     

The inhomogeneous strain distributions within finite cylinders under the axial point load tests and the effects on the valence band of Si1−xGex alloy

An exact solution for inhomogeneous strain and stress distributions within a finite cubic isotropic cylinder of Si_1?xGe_x alloy under the axial Point Load Strength Test (PLST) is analytically derived. Lekhnitskii’s stress function is used to uncouple the equations of equilibrium, and a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. The solution for isotropic cylinders under the axial PLST is covered as a special case. Numerical results show that the strain and stress distributions in the central region within half height and radius are relatively homogeneous, but strain and stress concentrations are usually developed near the point loads. The largest tensile strain and stress are always induced along the line joining the point loads, which gives theoretical explanation why most of the cylindrical specimens are split apart along the line joining the point loads under the axial PLST. In addition, by using envelope-function method, the effect of strain on the valence-band structure of Si_1?xGe_x alloy is analyzed. It is found that strain changes the band quantum gap and the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1?xGe_x alloy.     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

Evaluatin of the higashitani and pritchard analysis of the hole pressure using flow birefingence

Higashitani and Pritchard (H-P) carried out an analysis of the hole pressure (P_H) for viscoelastic fluids which leads to expressions relating P_H to the shear stress (σ), the wall shear stress (σ_w), and the normal stress differneces (N₁ and N₂). Although very good agreement their theory and experimental results has been obtained for several polymer solutions and three polymer melts, it is known that at least two of the key assumptions in the theory are violated. In this study flow birefingence has been used to determine the stress field (i.e. σ and the normal stress difference, σ₁₁ — σ₂₂) in the region of a slot placed perpendicular to the flow direction for a polystyrene melt. Values of σ and σ₁₁ — σ₂₂ were then used to evaluate the integrand in the expression relating P_e1 where P_e1 is the hole pressure measured at the base of the rectangular slot, σ and N₁. Values of P_e1 evaluated using flow birefringence data agreed well with those obtained using the same integral expression and cone-and-plate values of N₁ and σ, and with directly measured values of P_e1. This agreement occurred even though the stress field was found to be asymmetric around the centerline of the slot and with secondary flow in the slot. A detailed evaluation of the values of N₁/2σ, which constitute the integral in the H-P theory, along the centerline of the slot revealed most of the contributions to the integral canceled in integrating from the base of the slot where σ is zero at the centerline of the slit-die. The main contributions to the integral occurred from the integration taken from the centerline of the slit-die to the upperdie     

Finite element modeling of elasto-plastic contact between rough surfaces

This paper presents a finite element calculation of frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with a self-affine fractal surface. The calculations are conducted within an explicit dynamic Lagrangian framework. The elasto-plastic response of the material is described by a J₂ isotropic plasticity law. Parametric studies are used to establish general relations between contact properties and key material parameters. In all cases, the contact area A rises linearly with the applied load. The rate of increase grows as the yield stress σ_y decreases, scaling as a power of σ_y over the range typical of real materials. Results for A from different plasticity laws and surface morphologies can all be described by a simple scaling formula. Plasticity produces qualitative changes in the distributions of local pressures in the contact and of the size of connected contact regions. The probability of large local pressures is decreased, while large clusters become more likely. Loading-unloading cycles are considered and the total plastic work is found to be nearly constant over a wide range of yield stresses.