Nonaxisymmetric Deformation of Solids of Revolution Made of Elastic Orthotropic Materials with Different Tensile and Compressive Moduli

Savchenko, V. G.

doi:10.1007/s10778-005-0141-1

Int Appl Mech

2005

DOI: 10.1007/s10778-005-0141-1

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Nonaxisymmetric Deformation of Solids of Revolution Made of Elastic Orthotropic Materials with Different Tensile and Compressive Moduli

V. G. Savchenko

Abstract: A method is proposed to allow for the difference of the tensile and compressive moduli of compound orthotropic bodies of revolution subject to nonaxisymmetric loading and heating. The compliance matrix is symmetrized by introducing weighting coefficients that take into account the influence of the sign of stresses in two mutually perpendicular directions on the corresponding coefficients of this matrix Keywords: nonaxisymmetric thermostressed state, body of revolution, orthotropic materials, different tensile … Show more

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Cited by 7 publications

(10 citation statements)

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“…It allows doing structural analysis with regard to physical nonlinearity. However, the finite-element solution of spatial problems frequently involves significant technical difficulties associated with the implementation of efficient discretization algorithms; therefore, spatial objects are, as a rule, designed in a simplified formulation (axisymmetric or plane) [13][14][15][16]. The performance of the FEM can be improved by combining it with the variable separation method.…”

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confidence: 99%

On finite-element solution of spatial thermoviscoelastoplastic problems

Lelyukh

Shevchenko

2006

Int Appl Mech

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A finite-element algorithm is proposed for the analysis of the thermoviscoelastoplastic stress-strain state of bodies under complex loading (thermal and mechanical). It is assumed that an arbitrary element of the body deforms along a rectilinear or slightly curved path. The three-dimensional stress-strain state of the body's elements is determined using the iterative method of additional strains. The technique is tested by analyzing the three-dimensional viscoelastic stress-strain state of a hollow cylinder and the thermoplastic state of a disk Keywords: thermoviscoelastoplastic stress-strain state, finite-element technique, iterative method of additional strains, viscoelastic stress-strain state, hollow cylinder, thermoplastic state, diskThe finite-element method (FEM) is one of the most universal numerical methods for solving boundary-value problems. It allows doing structural analysis with regard to physical nonlinearity. However, the finite-element solution of spatial problems frequently involves significant technical difficulties associated with the implementation of efficient discretization algorithms; therefore, spatial objects are, as a rule, designed in a simplified formulation (axisymmetric or plane) [13][14][15][16]. The performance of the FEM can be improved by combining it with the variable separation method. This approach is called the semianalytic finite-element method (SFEM) [6]. Its application domain is limited to homogeneous bodies of revolution and prismatic bodies with hinged ends. The recently developed moment scheme of the SFEM has greatly expanded the range of problems solved by this method [1][2][3]12]. Moreover, the recent advances in computer technology make it possible to further develop the finite-element displacement method to solve nonlinear spatial problems of solid mechanics. When combined with efficient algorithms of successive approximations, such as different modifications of the Newton-Raphson method, this approach is distinguished by a relatively simple and clear procedure of deriving the discrete kinematic relations for unknown variables and the governing equations. Moreover, the spatial formulation of the problem imposes almost no restrictions on the geometry of the object of interest and on the type and spatial distribution of loads. The present paper addresses a finite-element technique of solving a three-dimensional thermoviscoelastoplastic stress-strain problem for three-dimensional bodies.1. Problem Formulation and Governing Equations. Consider, in an orthogonal curvilinear coordinates σ ϕϕ , an isotropic deformable solid of volume V bounded by a surface Σ. Suppose that at the initial point in time t = 0, the body, which is in natural stress-strain-free state at a temperature T 0 , begins to be subjected to nonuniform heating, body forces q i , and surface forces q i acting on a portion Σ t of the surface. The other portion Σ u of the surface can be fixed in a certain fashion, i.e., displacements r u can be prescribed on it. Suppose that the strain paths of an arbitrar...

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On finite-element solution of spatial thermoviscoelastoplastic problems

Lelyukh

Shevchenko

2006

Int Appl Mech

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“…In some fibrous and particulate materials, Young's moduli in tension (Å + ) and compression (Å -) may differ by even 50%. The tensile and compression stress-strain curves of such materials may also differ significantly.Methods to analyze the stress-strain state of structural members made of materials with different tensile and compressive moduli are outlined in [1,6,20]. Studies on damage of such materials are unavailable.…”

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confidence: 99%

“…Methods to analyze the stress-strain state of structural members made of materials with different tensile and compressive moduli are outlined in [1,6,20]. Studies on damage of such materials are unavailable.…”

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confidence: 99%

“…Though this approach has not been theoretically justified, it allows us to symmetrize the compliance matrix and to use anisotropic elasticity theory to solve the problem posed. With such an approach [20], the compliance matrix has the following elements, depending on the sign of stresses: | | | | …”

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confidence: 99%

“…Depending on the sign and magnitude of the stress components, determine the coefficients A ijkl and, hence, the functions A ijkl ijkl 0 w using formulas (1), (3)- (6). Set the parameters w ij D at zero in the first approximation of the first step of loading.…”

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Stress state of compound solids of revolution made of damaged orthotropic materials with different tensile and compressive moduli

Savchenko

2006

Int Appl Mech

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A technique is proposed to allow for damages and different tensile and compressive moduli of orthotropic materials in stress-strain analysis of compound bodies of revolution under nonaxisymmetric loading and heating. The technique combines the semi-analytic finite-element method and the method of successive approximations Keywords: nonaxisymmetric thermostressed state, solid of revolution, orthotropic materials with different tensile and compressive moduli, microdamageIntroduction. Many important engineering problems involve thermostress analysis of various structural members to infer their reliability, performance, and endurance. Efficient methods were developed in [3, 7, 11, etc.] to determine the thermoelastoplastic stress-strain state of bodies of various shapes. The accumulation of damages in viscoplastic isotropic materials is described by introducing a scalar parameter [2, 4, 10, 12, 14, 15, etc.]. This parameter is determined from a kinematic equation that relates its time derivative with some equivalent stress. Such an approach reduces the process of damage to loosening of a microvolume. Another approach to the description of damage is to use a structural model [9, 17, 18, etc.] based on stochastic equations for microinhomogeneous materials. Here, dispersed microdamages are modeled by quasispherical micropores that may be filled with particles of destroyed materials, and the accumulation of microdamages during deformation is modeled as increased porosity. Both approaches assume that all area elements associated with one point are equally damaged. Actually, the distribution of damages over a deformed element under combined stress was experimentally shown to be anisotropic. This can drastically affect the behavior of the material. Therefore, only one measure of damage accumulation may appear insufficient in many cases to correctly identify the onset of failure if for no other reason than different failure mechanisms provided by tangential and normal stresses. In this connection, the papers [5,19] propose to describe damages in orthotropic materials by introducing six damage parameters, one per each area element. These parameters account for changes in the initial structure of a material; nucleation, development, and coalescence of pores; and formation of microdefects during deformation, which decreases the effective area of sections over which the stress components are distributed. Also, these parameters help to explain the nonlinearity of tension, compression, torsion, and shear curves. A literature survey indicates that the numerical analysis of deformational damage in composite materials has received inadequate development.Although the curve of interatomic forces passes through zero smoothly, modern composite materials may have, at the macrolevel, different elastic moduli in tension and compression because of the presence of internal pores, inclusions, and other defects [1, 8, 13, 16, etc.]. In some fibrous and particulate materials, Young's moduli in tension (Å + ) and compression (Å -) may differ by...

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Linear relationship between the first invariants of the stress and strain tensors in theories of plasticity with strain hardening

2007

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Energy-coupled stress and strain measures are defined in Euler coordinates. They are used to analyze the relationship between the first invariants of the stress and strain tensors for linearity and to determine strains at which the plastic component of the first strain invariant can be neglected. It is established that this relationship remains linear within an engineering plastic-strain tolerance of 0.2% irrespective of the value of strain intensity, which depends on the type of material and its stress state Keywords: solid body, Eulerian and Lagrangian coordinates, stress and strain tensors, first invariants, linear relationshipIntroduction. The modern theories of plasticity with strain hardening [11][12][13][14][15], which refer to Bridgman's study [2], postulated the generalized Hooke's law in both elastic and plastic strain ranges; i.e., it is supposed that the volume of a solid does not change during elastoplastic deformation. In this case, the first invariant of the stress tensor is used as hydrostatic pressure and the first invariant of the strain tensor as volume strain; i.e., all modern theories of plasticity assume that the first invariants of the stress and strain tensors are in a linear relationship. However, the first strain invariant can define volume strain only approximately.In this connection, we will discuss the values of the strain components at which the plastic component of the first strain invariant can be neglected, which, in fact, is done in each modern theory of plasticity. We will use the principles of solid mechanics based on Cauchy's continuum hypothesis. Unlike Lagrange's approach, this hypothesis suggests using the method of sections to determine the stresses at an arbitrary point of a body on an area element somehow oriented in space and going through this point. According to this method, a solid subjected to specified external loads is partitioned by this plane, and one of the parts is rejected. The equilibrium condition for the remaining part is used to determine the principal vector and the principal moment exerted by the rejected part onto the remaining part in the specified section going through the specified point. Dividing the main vector and principal moment by the area of the section and letting it tend to zero at the point, we obtain that the limit of the ratio of the principal moment to this area tends to zero and the limit of the ratio of the principal vector to this area tends to the value of the stress vector acting on this plane at the specified point [3]. Projecting the vector onto the axes of an orthogonal coordinate system fixed at one point to the solid before deformation and having constant directions during deformation (Eulerian coordinate system), we obtain the stress components in a plane passing through the point of interest.1. In a deformed solid, we select an elementary rectangular parallelepiped with sizes dx i in the Eulerian coordinate system x i , apply the stresses obtained above to each of its sides, and set up differential equilibrium equations in t...

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Nonaxisymmetric Deformation of Solids of Revolution Made of Elastic Orthotropic Materials with Different Tensile and Compressive Moduli

Cited by 7 publications

References 6 publications

On finite-element solution of spatial thermoviscoelastoplastic problems

On finite-element solution of spatial thermoviscoelastoplastic problems

Stress state of compound solids of revolution made of damaged orthotropic materials with different tensile and compressive moduli

Linear relationship between the first invariants of the stress and strain tensors in theories of plasticity with strain hardening

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