One-axis stretching of the elasto-plastic plates with square hole of different orientation relatively the axis of stretching

На основі числового розв’язування крайових задач теорії малих пружно-пластичних деформацій для лінійно зміцнюваного матеріалу з урахуванням розвантаження з’ясовано вплив орієнтації квадратного отвору в пластині на її напружено-деформований стан за одновісного розтягу.Effective numerical methods for solving 2D problems related to the theories of elasticity and plasticity have been worked out. The variation-difference method of building finite difference schemes is extended to domains with curvilinear boundaries and disconnected domains. The application of the variation-difference method for solving problems of the theory of small elasto- plastic deformations relatively the plates with a hole, taking into account the linear strengthening of the material and unloading, has been developed. For solving the resultant systems of nonlinear and linear equation, the Newton-Kantorovich method and combined iterative method (gradient and cyclic Chebyshev’s one) were proposed to be used. The choice of iteration parameters of the methods for solving the obtained systems of linear and nonlinear algebraic equations was made. The elaborated software ensures solving the problems with different boundary conditions, medium and domain parameters. A variety of problems concerning one-axis stretching of the plates with the square hole of different orientation relatively the axis of stretching is numerically solved. The zones of evolution of plastic deformations for step enlarging of the loading are constructed, the remaining deformations and the limit loads are obtained. The influence of the orientation of square hole in the plate onto the stress magnitude was found. The orientation of the hole does not make essential influence onto the shape of zones of generation and development of residual deformations, but it essentially impacts onto magnitudes of the components of the stress tensor and its intensity, and therefore the moment of the beginning of plastic deformation. That is, in a plate with rotated square hole the initial plastic deformations appear under the tension 45% less than in a non-rotated one

Influence of stress concentrators onto stress-strain state of elasto-plastic plates

На основі числового розв’язування крайових задач теорії малих пружно-пластичних деформацій для лінійно зміцнюваного матеріалу з урахуванням розвантаження з’ясовано вплив концентраторів напружень (отвору, розрізу, абсолютно жорсткого включення) в пластині на її напружено-деформований стан за одновісного розтягу.Effective numerical methods for solving 2D problems related to the theories of elasticity and plasticity have been worked out. The variation-difference method of building finite difference schemes is extended to domains with curvilinear boundaries and disconnected domains. The application of the variation-difference method for solving problems of the theory of small elasto-plastic deformations relatively the plates with a hole and a cut, taking into account the linear strengthening of the material and unloading, has been developed. For solving the resultant systems of nonlinear and linear equation, the Newton-Kantorovich method and combined iterative method (gradient and cyclic Chebyshev’s one) were proposed to be used. The choice of iteration parameters of the methods for solving the obtained systems of linear and nonlinear algebraic equations was made. The elaborated software ensures solving the problems with different boundary conditions, medium and domain parameters. A variety of problems concerning one-axis stretching of the plates with a disk hole (or rigid body instead of a hole) and a cut (or a thin rigid body instead of a cut) is numerically solved. The zones of evolution of plastic deformations for step enlarging of the loading are constructed, the residual strains and the limit loads are obtained. The influence of a disk hole, cut and rigid body instead of a hole or cut in the plates onto the stress magnitude was found. The presence of rigid bodies instead of a hole or a cut strengthen the plate in general, in particular, plastic strains in the plate with rigid bodies instead of a hole and a cut appear under tension stress in 2,75 times more, than in the plate with a disk hole and a cut, and yield limit arrives at tension stress in 1,79 times more. A long thin rigid body strengthens a plate more than a disk one. The residual strains around a rigid body in the plates with a disk rigid body and a cut, and in the plates with a disk hole and a thin rigid body do not arise, in general

Stress-strain state of elasto-plastic linearly strengthenable plates with two perpendicular cuts under all-round stretching

За допомогою числового розв’язування крайових задач теорії малих пружно-пластичних деформацій для лінійно зміцнюваного матеріалу з’ясовано напружено-деформований стан пружно-пластичних пластин з двома перпендикулярними розрізами за всебічного розтягу.Effective numerical methods for solving 2D problems related to the theories of elasticity and plasticity have been worked out. The variation-difference method of building finite difference schemes is extended to disconnected domains. The application of the variation-difference method for solving problems of the theory of small elasto-plastic deformations relatively the plates with cuts, taking into account the linear strengthening of the material and unloading, has been developed. For solving the resultant systems of nonlinear and linear equation, the Newton-Kantorovich method and combined iterative method (gradient and cyclic Chebyshev’s one) were proposed to be used. The choice of iteration parameters of the methods for solving the obtained systems of linear and nonlinear algebraic equations was made. The elaborated software ensures solving the problems with different boundary conditions, medium and domain parameters. A variety of problems concerning all-round stretching of the elasto-plastic plates with two perpendicular cuts is numerically solved. The zones of evolution of plastic deformations for step enlarging of the loading are constructed. There are found the stresses under which the yield limit and the strength limit are achieve in the plates. On the base of numerical analysis the following main regularities are found: under the close mutual location of cuts in the plate, the plastic deformations first appear under the stress which is 33% less than in the plate with the same cuts under the far mutual location; however, the strength limit in the plates in the both considered cases is achieve practically under the same stress