Closed form solutions of the differential equations governing the plastic fracture field in a power-law hardening material with low strain-hardening exponent

1990 ◽  
Vol 60 (7) ◽  
pp. 444-462 ◽  
Author(s):  
D. E. Panayotounakos ◽  
M. Markakis
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Strain Hardening ◽  
Power Law ◽  
Strain Hardening Exponent ◽  
Hardening Material ◽  
Closed Form Solutions

Dynamic Spherical Cavity Expansion of Strain-Hardening Materials

1991 ◽  
Vol 58 (1) ◽  
pp. 1-6 ◽  
Author(s):  
V. K. Luk ◽  
M. J. Forrestal ◽  
D. E. Amos
Keyword(s):  
Numerical Solution ◽  
Closed Form ◽  
Strain Hardening ◽  
Power Law ◽  
Spherical Cavity ◽  
Incompressible Material ◽  
Cavity Expansion ◽  
Closed Form Solutions ◽  
Spherical Cavities ◽  
Spherical Cavity Expansion

We developed models for the dynamic expansion of spherical cavities from zero initial radii for elastic-plastic, rate-independent materials with power-law strain hardening. The models considered the material as incompressible and compressible. For an incompressible material, we obtained closed-form solutions, whereas the compressible results required the numerical solution of differential equations. A comparison of the numerical results from both models showed the effect of compressibility.

A Proposed Generalized Model for Forming Limit Diagram

2015 ◽  
Author(s):  
Aly El Domiaty ◽  
Abdel-Hamid I. Mourad ◽  
Abdel-Hakim Bouzid
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Strain Hardening ◽  
Power Law ◽  
Yield Criterion ◽  
Rate Of Change ◽  
Forming Limit ◽  
Strain Hardening Exponent ◽  
Sensitivity Factor ◽  
Generalized Model

One of the most significant approaches for predicting formability is the use of forming limit diagrams (FLDs). The development of the generalized model integrates other models. The first model is based on Von-Misses yield criterion (traditionally used for isotropic material) and power law constitutive equation considering the strain hardening exponent. The second model is also based on Von-Misses yield criterion but uses a power law constitutive equation that considers the effect of strain rate sensitivity factor. The third model is based on the modified Hill’s yield criterion (for anisotropic materials) and a power law constitutive equation that considers the strain hardening exponent. The current developed model is a generalized model which is formulated on the basis of the modified Hill yield criterion and a power law constitutive equation considering the effect of strain rate. A new controlling parameter (γ) for the limit strains was exploited. This parameter presents the rate of change of strain rate with respect to strain. As γ increases the level of the FLD raises indicating a better formability of the material.

A Simplified Model for Evaluating Strain Demand in a Pipeline Subjected to Longitudinal Ground Movement

2008 ◽  
Author(s):  
Nader Yoosef-Ghodsi ◽  
Joe Zhou ◽  
D. W. Murray
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Strain Hardening ◽  
Ground Movement ◽  
Simplified Model ◽  
Flow Rule ◽  
Pipe Material ◽  
Direction Parallel ◽  
Axial Forces ◽  
Sliding Zone

A simplified model was developed to calculate the maximum tensile and compressive strains due to a uniform movement of a block of soil in a direction parallel to the pipe axis using a closed-form solution of the governing differential equations. The model employs the theory of plasticity for modelling the pipe material based on normality plastic flow rule, the von Mises yield criterion, and isotropic strain hardening. While the pipe was assumed to have a bilinear, stress-strain curve with strain hardening, the pipe-soil friction was assumed to have an elastic-perfectly plastic force-deformation response. The model accounts for the initial thermal axial strains in the pipe and biaxial state of stress in the pipe due to internal pressure. The model is capable of accommodating pipe bends at the ends of the sliding zone. The relationship between the ground displacement and pipe axial force at each interface of stable and sliding zones was obtained from closed-form solutions of governing differential equations, assuming both the stable and sliding zones are infinitely long. To prevent the overestimation of the axial strains in the pipe, a limiting scenario was considered where the soil was assumed to have yielded over the entire sliding zone. Equilibrium and compatibility equations were used to calculate the pipe axial forces and strains at the two interfaces. The simplified model for longitudinal ground movement was validated against finite element solutions. The validation example presented involves a 20-inch straight pipeline subjected to longitudinal ground movement over slide lengths of 50, 100 and 200 metres, as well as a semi-infinite sliding zone case.

Shakedown Loads for Pressure Vessels of Strain Hardening Materials

1969 ◽  
Vol 11 (3) ◽  
pp. 340-342 ◽  
Author(s):  
T. E. Taylor
Keyword(s):  
Strain Hardening ◽  
Power Law ◽  
Pressure Vessels ◽  
Strain Hardening Exponent ◽  
Rigid Plastic ◽  
Linear Elastic ◽  
Perfectly Plastic ◽  
Creep Analysis ◽  
Family Of Curves ◽  
The Relationship

A power law, well known in creep analysis, embodies a family of curves which express the stress-strain relations for a family of materials ranging from linear elastic to rigid perfectly plastic. A linearization of the relationship between stress concentration factor and the reciprocal of strain hardening exponent for geometrically similar pressure vessels made of materials within the family has enabled a view of shakedown in vessels of strain hardening materials to be formulated. The absence of discontinuities in the power law, except at the rigid plastic end point, results in shakedown loads dependent on strain hardening exponent and previous loading history.

THREE-DIMENSIONAL SIMULATION OF DEFORMATION FIELDS UNDERNEATH VICKERS INDENTER: EFFECTS OF POWER-LAW PLASTICITY

2010 ◽  
Vol 24 (01n02) ◽  
pp. 238-246 ◽  
Author(s):  
NUWONG CHOLLACOOP ◽  
UPADRASTA RAMAMURTY
Keyword(s):  
Yield Strength ◽  
Strain Hardening ◽  
Power Law ◽  
Strain Distribution ◽  
Three Dimensional ◽  
Strain Hardening Exponent ◽  
Al Alloy ◽  
Element Analysis ◽  
Vickers Indenter ◽  
Experimental Heat

The effects of power-law plasticity (yield strength and strain hardening exponent) on the plastic strain distribution underneath a Vickers indenter was systematically investigated by recourse to three-dimensional finite element analysis, motivated by the experimental macro- and micro-indentation on heat-treated Al - Zn - Mg alloy. For meaningful comparison between simulated and experimental results, the experimental heat treatment was carefully designed such that Al alloy achieve similar yield strength with different strain hardening exponent, and vice versa. On the other hand, full 3D simulation of Vickers indentation was conducted to capture subsurface strain distribution. Subtle differences and similarities were discussed based on the strain field shape, size and magnitude for the isolated effect of yield strength and strain hardening exponent.

scholarly journals Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 226
Author(s):  
Efthimios Providas ◽  
Stefanos Zaoutsos ◽  
Ioannis Faraslis
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Closed Form ◽  
Linear Operators ◽  
General Boundary ◽  
Order Differential Operator ◽  
General Boundary Conditions ◽  
Symbolic Form ◽  
Closed Form Solutions

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces.

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