scholarly journals Qualitative Behaviour of Elastic-Plastic Solutions for a Class of Damage Mechanics Models Near Bimaterial Interfaces: A Simple Analytical Example

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Sergei Alexandrov ◽  
Yusof Mustafa
Keyword(s):  
Plastic Strain ◽  
Boundary Value Problem ◽  
Damage Mechanics ◽  
Boundary Value ◽  
Equivalent Plastic Strain ◽  
Damage Parameter ◽  
Bimaterial Interface ◽  
Hard Material ◽  
Qualitative Behaviour ◽  
Elastic Plastic

The paper presents an exact analytic solution for a class of elastic-plastic models with damage evolution. The boundary value problem consists of a planar deformation comprising the simultaneous shearing and expansion of a hollow cylindrical specimen of material and involves a bimaterial interface at which the materials stick to each other. With no loss of generality for understanding the qualitative behaviour of the solution near the bimaterial interface, an extreme case when the hard material is rigid is considered. The solution is reduced to a transcendental equation for the value of the equivalent plastic strain at the bimaterial interface. This equation predicts that the equivalent plastic strain attains a maximum under certain conditions. The existence of the solution of the boundary value problem depends on the value of the damage parameter at fracture, which is a material constant. In particular, if this value is larger than the value of the damage parameter at the bimaterial interface corresponding to the maximum possible value of the equivalent strain at this interface, then no solution exists. Experimental data available in the literature are used to assess whether Lemaitre’s model is applicable.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

Solid Formulation, Comparison of Two Formulations, and Concluding Remarks

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Finite Element ◽  
Field Equation ◽  
Boundary Value Problem ◽  
Metal Forming ◽  
Large Strain ◽  
Boundary Value ◽  
Elastic Plastic ◽  
Plastic Materials ◽  
Elastic Plastic Materials ◽  
Selection Of

In the previous chapters we have discussed only the applications of flow formulation to the analysis of metal-forming processes. Lately, elastic-plastic (solid) formulations have evolved to produce techniques suitable for metal-forming analysis. This evolution is the result of developments achieved in large-strain formulation, beginning from the infinitesimal approach based on the Prandtl–Reuss equation. A question always arises as to the selection of the approach—“flow” approach or “solid” approach. A significant contribution to the solution of this question was made through a project in 1978, coordinated by Kudo, in which an attempt was made to examine the comparative merits of various numerical methods. The results were compiled for upsetting of circular solid cylinders under specific conditions, and revealed the importance of certain parameters used in computation, such as mesh systems and the size of an increment in displacement. This project also showed that the solid formulation needed improvement, particularly in terms of predicting the phenomenon of folding. For elastic-plastic materials, the constitutive equations relate strain–rate to stress–rates, instead of to stresses. Consequently, it is convenient to write the field equation in the boundary-value problem for elastic-plastic materials in terms of the equilibrium of stress rates. In this chapter, the basic equations for the finite-element discretization involved in solid formulations are outlined both for the infinitesimal approach and for large-strain theory. Further, the solutions obtained by the solid formulation are compared with those obtained by the flow formulation for the problems of plate bending and ring compression. A discussion is also given concerning the selection of the approach for the analysis. In conclusion, significant recent developments in the role of the finite-element method in metal-forming technology are summarized. The field equation for the boundary-value problem associated with the deformation of elastic-plastic materials is the equilibrium equation of stress rates. As stated in Chap. 1 (Section 1.3), the internal distribution of stress, in addition to the current states of the body, is supposed to be known, and the boundary conditions are prescribed in terms of velocity and traction-rate.

Effects of Local Peak Stress Distribution on the Ratchet Limit

2002 ◽  
Author(s):  
Nobuyoshi Yanagida ◽  
Masaaki Tanaka ◽  
Norimichi Yamashita ◽  
Yukinori Yamamoto
Keyword(s):  
Finite Element ◽  
Stress Distribution ◽  
Plastic Strain ◽  
Equivalent Stress ◽  
Equivalent Plastic Strain ◽  
Peak Stress ◽  
Element Analysis ◽  
Stress Evaluation ◽  
Elastic Plastic ◽  
Plastic Analysis

Alternative stress evaluation criteria suitable for Finite Element Analysis (FEA) proposed by Okamoto et al. [1],[2] have been studied by the Committee on Three Dimensional Finite Element Stress Evaluation (C-TDF) in Japan. Thermal stress ratchet criteria in plastic FEA are now under consideration. Two criteria are proposed: (1) Evaluating variations in plastic strain increments, and (2) Evaluating the width of the area in which Mises equivalent stress exceeds 3Sm. To verify of these criteria, we selected notched cylindrical vessel models as prime elements. To evaluate the effect of the local peak stress distribution on these criteria, cylindrical vessels with a semicircular notch on the outer surface were selected for this analysis. We used two notch configurations for our analysis, and the stress concentration factor for the notches was set to 1.5 and 2.0. We conducted elastic-plastic analysis to evaluate the ratchet limit. Sustained pressure and alternating enforced longitudinal displacements which causes secondary stress were used as parameters for the elastic-plastic analysis. We found that when no ratchet was observed, the equivalent plastic strain increments decreased and the area in which Mises equivalent stress exceeds 3Sm are below the certain range.

Solution of the Displacement Boundary Value Problem of an Interface Between Two Dissimilar Half-Planes and a Rigid Elliptic Inclusion at the Interface

1998 ◽  
Vol 65 (4) ◽  
pp. 880-888 ◽  
Author(s):  
V. Boniface ◽  
N. Hasebe
Keyword(s):  
Stress Concentration ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Function Technique ◽  
Bimaterial Interface ◽  
Radius Of Curvature ◽  
Elliptic Inclusion ◽  
Displacement Boundary ◽  
Concentration Factors ◽  
Stress Concentration Factors

The displacement boundary value problem of a bimaterial interface is solved using the complex stress function technique. A rational mapping function is used to map the two half-planes into unit circles and analysis is carried out in the mapped plane. The symmetric bimaterial problem is considered and the particular case of a rigid elliptic inclusion at the interface is solved. Uniform remote tensions both along and normal to the interface are considered. Stress distributions on the inclusion boundary are shown. Stress concentration factors at the inclusion tips are obtained and are expressed in terms of the radius of curvature using an approximate form of a general expression. These results are used to predict the likelihood of debonding/cracking at the tips. Also, stress concentration factors at the tips of an elliptic inclusion and elliptic void are compared. Stress intensity factors at the tips of a thin rigid elliptic inclusion are also determined.

scholarly journals A Coupled Elastoplastic Damage Dynamic Model for Rock

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xuelong Hu ◽  
Ming Zhang ◽  
Xiangyang Zhang ◽  
Min Tu ◽  
Zhiqiang Yin ◽  
...  
Keyword(s):  
Plastic Strain ◽  
Constitutive Model ◽  
Damage Mechanics ◽  
Yield Criterion ◽  
Triaxial Compression ◽  
Equivalent Plastic Strain ◽  
Damage Variable ◽  
Uniaxial Tensile ◽  
Uniaxial Tensile Test ◽  
Elastoplastic Damage

Rock dynamic constitutive model plays an important role in understanding dynamic response and addressing rock dynamic problems. Based on elastoplastic mechanics and damage mechanics, a dynamic constitutive model of rock coupled with elastoplastic damage is established. In this model, unified strength theory is taken as the yield criterion; to reflect the different damage evolution law of rocks under tension and pressure conditions, the effective plastic strain and volumetric plastic strain are used to represent the compressive damage variable and the equivalent plastic strain is used to represent the tensile damage variable; the plastic hardening behavior and strain rate effect of rocks are characterized by piecewise function and dynamic increase factor function, respectively; Fortran language and LS-DYNA User-Defined Interface (Umat) are used to numerically implement the constitutive model; the constitutive model is verified by three classical examples of rock uniaxial and triaxial compression tests, rock uniaxial tensile test, and rock ballistic test. The results show that the constitutive model can describe the dynamic and static mechanical behavior of rock comprehensively.

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