Velocity of an elastic unloading wave in steel

1986 ◽  
Vol 18 (9) ◽  
pp. 1225-1228
Author(s):  
G. V. Stepanov ◽  
V. I. Zubov
Keyword(s):  
Unloading Wave ◽  
Elastic Unloading

Wave propagation in shape memory alloy rods under impulsive loads

2005 ◽  
Vol 461 (2064) ◽  
pp. 3871-3892 ◽  
Author(s):  
Yi-Chao Chen ◽  
Dimitris C Lagoudas
Keyword(s):  
Wave Propagation ◽  
Shape Memory Alloy ◽  
Shape Memory ◽  
Wave Front ◽  
Impact Load ◽  
Reverse Transformation ◽  
Loading Function ◽  
Unloading Wave ◽  
The Impact ◽  
Elastic Unloading

A dynamic analysis is given of the wave propagation in a polycrystalline shape memory alloy semi-infinite rod subject to an impulsive load described by a rectangular loading function. This is a continuation of the previous study for which the impact load is described by a step loading function, and for which the rod undergoes martensitic phase transformation only. In the presence of unloading, the rod also undergoes reverse transformation from martensite to austenite. A maximum internal dissipation criterion is proposed to select a unique dynamic solution, in which an elastic unloading wave front precedes the reverse transformation wave front. This elastic unloading wave will meet with the forward transformation wave at a later time, leading to a sequence of wave encounters and progressively more complicated profiles of the state variables. The asymptotic behaviour of the solution and the energy dissipation are discussed.

scholarly journals Finite Plane Strain Bending under Tension of Isotropic and Kinematic Hardening Sheets

Materials ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1166
Author(s):  
Stanislav Strashnov ◽  
Sergei Alexandrov ◽  
Lihui Lang
Keyword(s):  
Plane Strain ◽  
Kinematic Hardening ◽  
Tensile Force ◽  
Plastic Region ◽  
Equivalent Plastic Strain ◽  
Isotropic Hardening ◽  
Finite Plane ◽  
Present Solution ◽  
Bending Under Tension ◽  
Elastic Unloading

The present paper provides a semianalytic solution for finite plane strain bending under tension of an incompressible elastic/plastic sheet using a material model that combines isotropic and kinematic hardening. A numerical treatment is only necessary to solve transcendental equations and evaluate ordinary integrals. An arbitrary function of the equivalent plastic strain controls isotropic hardening, and Prager’s law describes kinematic hardening. In general, the sheet consists of one elastic and two plastic regions. The solution is valid if the size of each plastic region increases. Parameters involved in the constitutive equations determine which of the plastic regions reaches its maximum size. The thickness of the elastic region is quite narrow when the present solution breaks down. Elastic unloading is also considered. A numerical example illustrates the general solution assuming that the tensile force is given, including pure bending as a particular case. This numerical solution demonstrates a significant effect of the parameter involved in Prager’s law on the bending moment and the distribution of stresses at loading, but a small effect on the distribution of residual stresses after unloading. This parameter also affects the range of validity of the solution that predicts purely elastic unloading.

Improved Compliance Solutions for C(T), SE(B) and Clamped SE(T) Specimens Including Side-Grooves, Varying Thicknesses and 3-D Effects

2014 ◽  
Author(s):  
Gustavo Henrique B. Donato ◽  
Felipe Cavalheiro Moreira
Keyword(s):  
Crack Growth ◽  
Plane Strain ◽  
Plane Stress ◽  
Potential Drop ◽  
Crack Size ◽  
Growth Conditions ◽  
Size Estimation ◽  
Unloading Compliance ◽  
Integrity Assessments ◽  
Elastic Unloading

Fracture toughness and Fatigue Crack Growth (FCG) experimental data represent the basis for accurate designs and integrity assessments of components containing crack-like defects. Considering ductile and high toughness structural materials, crack growing curves (e.g. J-R curves) and FCG data (in terms of da/dN vs. ΔK or ΔJ) assumed paramount relevance since characterize, respectively, ductile fracture and cyclic crack growth conditions. In common, these two types of mechanical properties severely depend on real-time and precise crack size estimations during laboratory testing. Optical, electric potential drop or (most commonly) elastic unloading compliance (C) techniques can be employed. In the latter method, crack size estimation derives from C using a dimensionless parameter (μ) which incorporates specimen’s thickness (B), elasticity (E) and compliance itself. Plane stress and plane strain solutions for μ are available in several standards regarding C(T), SE(B) and M(T) specimens, among others. Current challenges include: i) real specimens are in neither plane stress nor plane strain - modulus vary between E (plane stress) and E/(1-ν2) (plane strain), revealing effects of thickness and 3-D configurations; ii) furthermore, side-grooves affect specimen’s stiffness, leading to an “effective thickness”. Previous results from current authors revealed deviations larger than 10% in crack size estimations following existing practices, especially for shallow cracks and side-grooved samples. In addition, compliance solutions for the emerging clamped SE(T) specimens are not yet standardized. As a step in this direction, this work investigates 3-D, thickness and side-groove effects on compliance solutions applicable to C(T), SE(B) and clamped SE(T) specimens. Refined 3-D elastic FE-models provide Load-CMOD evolutions. The analysis matrix includes crack depths between a/W=0.1 and a/W=0.7 and varying thicknesses (W/B = 4, W/B = 2 and W/B = 1). Side-grooves of 5%, 10% and 20% are also considered. The results include compliance solutions incorporating all aforementioned effects to provide accurate crack size estimation during laboratory fracture and FCG testing. All proposals revealed reduced deviations if compared to existing solutions.

scholarly journals Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 145
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Yeong-Maw Hwang
Keyword(s):  
Plane Strain ◽  
Yield Criterion ◽  
Large Strains ◽  
Rigid Plastic ◽  
Plastic Properties ◽  
Elastic Plastic ◽  
Starting Point ◽  
The Difference ◽  
Inside Radius ◽  
Elastic Unloading

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

Elastic Waves Created During Tensile Fracture: The Phenomenon of a Second Fracture

1953 ◽  
Vol 20 (1) ◽  
pp. 122-130
Author(s):  
Julius Miklowitz
Keyword(s):  
Cross Sections ◽  
Tensile Tests ◽  
Analog Computer ◽  
Tensile Fracture ◽  
Strain Waves ◽  
Unloading Wave ◽  
Second Fracture ◽  
Wave Compression ◽  
Initial Fracture ◽  
The Moment

Abstract In some tensile tests with brittle materials, it was noted that fractures were produced at two different cross sections of the specimen when the rupture load was reached. The phenomenon of the second fracture prompted the present investigation. It is believed that the second fracture is caused by the destructive action of the elastic strain waves created during the first of the two fractures. The analytical and experimental work carried out was focused on describing the character of these waves. Consideration of the mechanics involved reduces the problem to that of a vibrating cantilever beam with time-dependent boundary conditions. Two types of waves are shown to exist. The first is a longitudinal unloading wave (compression). The other is a group of flexural strain waves caused by the moment that develops at the initial fracture section. The methods of operational mathematics and the electric-analog computer have been employed in the analytical study.

The Asymptotic Elastic-Viscoplastic Field at Mode I Dynamic Propagating Crack-Tip

2007 ◽  
Vol 348-349 ◽  
pp. 817-820
Author(s):  
Zhen Qing Wang ◽  
Ji Bin Wang ◽  
Wen Yan Liang ◽  
Juan Su
Keyword(s):  
Crack Tip ◽  
Viscosity Coefficient ◽  
Plastic Strain Rate ◽  
Mode I ◽  
Mode I Crack ◽  
Moving Crack ◽  
Dynamic Propagating ◽  
Propagating Crack ◽  
Hardening Condition ◽  
Elastic Unloading

The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to the power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis is carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution is obtained containing no discontinuities. The variations of the numerical solution are discussed for mode I crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at the crack-tip only can be matched reasonably under linear-hardening condition. The tip field contains no elastic unloading zone for mode I crack.

scholarly journals Influence of the Replacement of the Actual Plastic Orthotropy with Various Approximations of Normal Anisotropy on Residual Stresses and Strains in a Thin Disk Subjected to External Pressure

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1834
Author(s):  
Yaroslav Erisov ◽  
Sergei Surudin ◽  
Sergei Alexandrov ◽  
Lihui Lang
Keyword(s):  
Yield Criterion ◽  
Plastic Anisotropy ◽  
Practical Importance ◽  
External Pressure ◽  
Metallic Materials ◽  
Normal Anisotropy ◽  
Residual Strains ◽  
Axis Of Symmetry ◽  
Residual Stresses And Strains ◽  
Elastic Unloading

Plastic anisotropy is very common to metallic materials. This property may significantly affect the performance of structures. However, the actual orthotropic yield criterion is often replaced with a criterion based on the assumption of normal anisotropy. The present paper aims to reveal the influence of this replacement on the distribution of strains and residual strains in a thin hollow disk under plane stress conditions. The boundary-value problem is intentionally formulated such that it is possible to obtain an exact semi-analytical solution without relaxing the boundary conditions. It is assumed that the disk is loaded by external pressure, followed by elastic unloading. The comparative analysis of the distributions of residual strains shows a significant deviation of the distribution resulting from the solutions based on the assumption of normal anisotropy from the distribution found using the actual orthotropic yield criterion. This finding shows that replacing the actual orthotropic yield criterion with the assumption of normal anisotropy may result in very inaccurate predictions. The type of anisotropy accepted is of practical importance because it usually results from such processes as drawing end extrusion with an axis of symmetry.

The Bauschinger and Hardening Effect on Residual Stresses in an Autofrettaged Thick-Walled Cylinder

1986 ◽  
Vol 108 (1) ◽  
pp. 108-112 ◽  
Author(s):  
P. C. T. Chen
Keyword(s):  
Residual Stress ◽  
Residual Stresses ◽  
Closed Form Solution ◽  
High Strength ◽  
Form Solution ◽  
Residual Stress Distribution ◽  
Hardening Effect ◽  
Reverse Yielding ◽  
Walled Cylinder ◽  
Elastic Unloading

Most of the earlier solutions for residual stresses were based on the assumption of elastic unloading and only a few considered reverse yielding. In this paper a new theoretical model for a high strength steel is proposed and a closed-form solution of residual stresses in autofrettaged tubes has been obtained. The new results indicate that the influence of the combined Bauschinger and hardening effect on the residual stress distribution is significant.

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