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Figure 13. Notched specimen subjected to cyclic loading: geometry and mesh.

Figure 13 Notched specimen subjected to cyclic loading: geometry and mesh.

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where B is the, as yet undetermined, assumed strain—displacement operator and » = 6éu is the test function. Consequently, we can write the weak problem of finding u and B that satisfy the balance residual where the test function du is designed to vanish along the parts of the boundary dQ, (where essen- tial boundary conditions are prescribed). Note that both conditions are now expressed in terms of the unknown displacement field, operated upon by both the symmetric gradient and the assumed symmetric gradient: the role of the assumed strain is now carried by the (at this point undetermined) assumed strain—displacement gradient operator B.
where B is the, as yet undetermined, assumed strain—displacement operator and » = 6éu is the test function. Consequently, we can write the weak problem of finding u and B that satisfy the balance residual where the test function du is designed to vanish along the parts of the boundary dQ, (where essen- tial boundary conditions are prescribed). Note that both conditions are now expressed in terms of the unknown displacement field, operated upon by both the symmetric gradient and the assumed symmetric gradient: the role of the assumed strain is now carried by the (at this point undetermined) assumed strain—displacement gradient operator B.
Table I. NICE-H8 element configuration. for fixed J and J € nodes(elems(/)). Because we are interested in the strain—displacement operators at the integration points (at the nodes, in our case), we will reverse the order of the sums
Table I. NICE-H8 element configuration. for fixed J and J € nodes(elems(/)). Because we are interested in the strain—displacement operators at the integration points (at the nodes, in our case), we will reverse the order of the sums
Ce ee nee en nn SIE I I * The matrix B; and the Jacobian 7(xx) are multi-valued at the node K. On the other hand, the constitutive matrix ID(xx) should be assumed to be single-valued. If this was not the case, we would be computing the strain—displacement operator across a material interface, and that must be avoided because of the expected jump in strains. Finally, By and By are independent of the index e. Therefore, rewriting (15) as This finally gives the definition of the assumed strain nodal matrix as a weighted average of the elementwise strain—displacement matrices
Ce ee nee en nn SIE I I * The matrix B; and the Jacobian 7(xx) are multi-valued at the node K. On the other hand, the constitutive matrix ID(xx) should be assumed to be single-valued. If this was not the case, we would be computing the strain—displacement operator across a material interface, and that must be avoided because of the expected jump in strains. Finally, By and By are independent of the index e. Therefore, rewriting (15) as This finally gives the definition of the assumed strain nodal matrix as a weighted average of the elementwise strain—displacement matrices
As an example, consider the mesh shown in Figure 1. The assumed strain—displacement matrices calculated at the quadrature point located at node 14 are going to be nonzero only for six nodes: 5, 11, 15, 13, 17, and 23. These matrices are evaluated from Equation (18) as Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2014; 99:844-866 DOI: 10.1002/nme It can be noted that the averaged strain matrix is constructed over the patch by means of standard objects and so the basic structure of the standard finite element algorithm is not very far from the present algorithm. The implementation of the present technique requires to know which elements are sharing the J node, and then the computation of B, is straightforward.
As an example, consider the mesh shown in Figure 1. The assumed strain—displacement matrices calculated at the quadrature point located at node 14 are going to be nonzero only for six nodes: 5, 11, 15, 13, 17, and 23. These matrices are evaluated from Equation (18) as Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2014; 99:844-866 DOI: 10.1002/nme It can be noted that the averaged strain matrix is constructed over the patch by means of standard objects and so the basic structure of the standard finite element algorithm is not very far from the present algorithm. The implementation of the present technique requires to know which elements are sharing the J node, and then the computation of B, is straightforward.
Figure 1. Regular rectangular grid, the center node is 14. 3. MIDPOINT INTEGRATION ALGORITHM FOR VON MISES PLASTICITY 3.1. Constitutive equations We consider classical small strain von Mises plasticity with linear isotropic hardening and both linear and nonlinear kinematic hardening mechanisms [27]. Splitting stress and strain tensors, 0 and €, into deviatoric and volumetric parts gives
Figure 1. Regular rectangular grid, the center node is 14. 3. MIDPOINT INTEGRATION ALGORITHM FOR VON MISES PLASTICITY 3.1. Constitutive equations We consider classical small strain von Mises plasticity with linear isotropic hardening and both linear and nonlinear kinematic hardening mechanisms [27]. Splitting stress and strain tensors, 0 and €, into deviatoric and volumetric parts gives
Figure 2. Stretched plate with circular hole. A perforated strip in uniaxial tension under plane stress condition is considered. This problem is a popular benchmark used to assess the precision and the effectiveness of the numerical technique
Figure 2. Stretched plate with circular hole. A perforated strip in uniaxial tension under plane stress condition is considered. This problem is a popular benchmark used to assess the precision and the effectiveness of the numerical technique
Table II. Energy norm values. [36-40]. The perforated plate is assumed to have the following material properties: Young’s modulus, E = 7000 Kg/mm?, Poisson’s ratio, v = 0.2; and yield stress, 0, = 24.3 Kg/mm?. With reference to Figure 2(a), the following dimensions are considered: W = 20, L = 36, plate thickness Figure 3. Stretched plate with circular hole: load—deflection curves.
Table II. Energy norm values. [36-40]. The perforated plate is assumed to have the following material properties: Young’s modulus, E = 7000 Kg/mm?, Poisson’s ratio, v = 0.2; and yield stress, 0, = 24.3 Kg/mm?. With reference to Figure 2(a), the following dimensions are considered: W = 20, L = 36, plate thickness Figure 3. Stretched plate with circular hole: load—deflection curves.
Figure 4. Stretched plate with circular hole—Von Mises comparison: H8 element uses SPR recovery; C3D8H element uses continuous plot option; and NICE-H8 element is used without post-processing operation. Minimum and maximum values are reported below each plot. A ne Table TT reports on the iterative energy norms at each loading step, computed with (46), showing yptimal convergence rate. Interestingly, the number of iterations required to meet the convergence riteria is stable throughout the loading path, namely, even when substantial plastic deformatior yecurs. The load deflection curves reported in Figure 3, given in terms of nodal reaction sum against he imposed displacement, indicate a perfect agreement between the present solution and different eference solutions. In Figures 4 and 5, the Von Mises stress plot and the equivalent plastic strain ure given, respectively. NICE-H8 recovered field results are compared with those provided by the 18 element, post-processed by the SPR [41, 42] technique, and the C3D8H element that uses uni- ‘orm interval type that creates a uniform, arithmetic progression between the interval values [34]
Figure 4. Stretched plate with circular hole—Von Mises comparison: H8 element uses SPR recovery; C3D8H element uses continuous plot option; and NICE-H8 element is used without post-processing operation. Minimum and maximum values are reported below each plot. A ne Table TT reports on the iterative energy norms at each loading step, computed with (46), showing yptimal convergence rate. Interestingly, the number of iterations required to meet the convergence riteria is stable throughout the loading path, namely, even when substantial plastic deformatior yecurs. The load deflection curves reported in Figure 3, given in terms of nodal reaction sum against he imposed displacement, indicate a perfect agreement between the present solution and different eference solutions. In Figures 4 and 5, the Von Mises stress plot and the equivalent plastic strain ure given, respectively. NICE-H8 recovered field results are compared with those provided by the 18 element, post-processed by the SPR [41, 42] technique, and the C3D8H element that uses uni- ‘orm interval type that creates a uniform, arithmetic progression between the interval values [34]
Figure 5. Stretched plate with circular hole—equivalent plastic strain: H8 element uses SPR recovery; C3D8H element uses continuous plot option; and NICE-H8 element is used without post-processing operation. Maximum value is reported below each plot.
Figure 5. Stretched plate with circular hole—equivalent plastic strain: H8 element uses SPR recovery; C3D8H element uses continuous plot option; and NICE-H8 element is used without post-processing operation. Maximum value is reported below each plot.
Figure 6. Cook-tapered beam. This problems configuration is shown in Figure 6. Young’s modulus is E = 70.0, the yield stress is oy = 0.243, and the incompressibility condition is enforced by setting Poisson’s ratio to
Figure 6. Cook-tapered beam. This problems configuration is shown in Figure 6. Young’s modulus is E = 70.0, the yield stress is oy = 0.243, and the incompressibility condition is enforced by setting Poisson’s ratio to
Figure 7. Cook-tapered beam: (a) equivalent plastic strain and (b) Von Mises stress with NICE-H8 ele- ment. Results obtained for a 10 x 10 mesh elements without any post-processing operation. Minimum and maximum values are reported below each plot.
Figure 7. Cook-tapered beam: (a) equivalent plastic strain and (b) Von Mises stress with NICE-H8 ele- ment. Results obtained for a 10 x 10 mesh elements without any post-processing operation. Minimum and maximum values are reported below each plot.
Figure 8. Cook-tapered beam: (a) equivalent plastic strain and (b) Von Mises stress with NICE-H8 element. Results obtained for a 20 x 20 mesh elements without any post-processing operation. Minimum and maximum values are reported below each plot. ~ _— As it can be noted from the figure, small differences in terms of measured displacements at the tip of the beam are obtained. Results obtained for the 10 x 10 mesh and the 20 x 20 mesh are in very good agreement. Note that the NICE element, due to its averaged patch formulation, tends to spread out singularities, strains, and stress concentrations. Nevertheless, accurate recovered field are obtained also for simple coarse mesh: note the narrow area with zero plastic strain values at the center of the beam, Figure 7 in comparison with 8.
Figure 8. Cook-tapered beam: (a) equivalent plastic strain and (b) Von Mises stress with NICE-H8 element. Results obtained for a 20 x 20 mesh elements without any post-processing operation. Minimum and maximum values are reported below each plot. ~ _— As it can be noted from the figure, small differences in terms of measured displacements at the tip of the beam are obtained. Results obtained for the 10 x 10 mesh and the 20 x 20 mesh are in very good agreement. Note that the NICE element, due to its averaged patch formulation, tends to spread out singularities, strains, and stress concentrations. Nevertheless, accurate recovered field are obtained also for simple coarse mesh: note the narrow area with zero plastic strain values at the center of the beam, Figure 7 in comparison with 8.
Figure 9. Cook-tapered beam: displacement at the tip of the beam versus applied force.
Figure 9. Cook-tapered beam: displacement at the tip of the beam versus applied force.
Figure 10. Thick cylinder example: geometrical dimensions and boundary conditions. Thick cylinder pressure—displacement curves are given in Figure 11(a), whereas hoop stress dis- tribution when plastic zone extends to 1.67;, is given in Figure 11(b). Stresses show an excellent comparison with the results obtained using the C3D8H element and the reference solution [45]. In this problem, the effect of varying the number of elements along the radius was investigated. The results are remarkable as no significant differences in displacements are obtained between the solutions using 5, 10, and 20 elements along the radius.
Figure 10. Thick cylinder example: geometrical dimensions and boundary conditions. Thick cylinder pressure—displacement curves are given in Figure 11(a), whereas hoop stress dis- tribution when plastic zone extends to 1.67;, is given in Figure 11(b). Stresses show an excellent comparison with the results obtained using the C3D8H element and the reference solution [45]. In this problem, the effect of varying the number of elements along the radius was investigated. The results are remarkable as no significant differences in displacements are obtained between the solutions using 5, 10, and 20 elements along the radius.
Figure 11. Thick cylinder example: (a) pressure—displacement curves, (b) hoop stress distribution along A-B when plastic zone extends to 1.6r;,. NICE results are plotted using a mesh with 5 x 9 (e) and a mesh with 10 x 18 elements (0) along the radius and circumference, respectively.
Figure 11. Thick cylinder example: (a) pressure—displacement curves, (b) hoop stress distribution along A-B when plastic zone extends to 1.6r;,. NICE results are plotted using a mesh with 5 x 9 (e) and a mesh with 10 x 18 elements (0) along the radius and circumference, respectively.
Figure 12. Thick cylinder example: Von Mises and equivalent plastic strain field recovered. Min and max values are reported for each computed quantity. NICE and ABAQUS results are plotted when plastic zone extends to 1.67;,. Note that NICE recovered quantities are not post-processed. In Figure 12, the Von Mises stress plot and the equivalent plastic strain are illustrated. NICE-H8 recovered field results are compared with those provided by the C3D8H element illustrated by means of the uniform interval type [34].
Figure 12. Thick cylinder example: Von Mises and equivalent plastic strain field recovered. Min and max values are reported for each computed quantity. NICE and ABAQUS results are plotted when plastic zone extends to 1.67;,. Note that NICE recovered quantities are not post-processed. In Figure 12, the Von Mises stress plot and the equivalent plastic strain are illustrated. NICE-H8 recovered field results are compared with those provided by the C3D8H element illustrated by means of the uniform interval type [34].
Figure 14. Notched specimen subjected to cyclic loading: cyclic response for purely kinematic hardening (solid line) and for purely isotropic hardening (dashed line). A circumferentially notched bar of circular cross section under axial loading cycles is considered [46]. The notched specimen is a shaft whose net section diameter measures 25.4 units and the cir- cumferential notch has a radius of 12.7 units, see Figures 15 and 16 for geometry and mesh details. Kinematic hardening model is used along with the following material properties: E = 2.1 x 10°, v = 0.3, oyo = 250, Hiso = 0, Akin = 1.1 x 10°, and Hy; = 1.0 x 103. Owing to the sym- metry of the problem, only one-eighth of the billet is modeled. A prescribed displacement 6 = 0.2 unit is applied to the lateral boundaries on the boundary surface with trace CD, whereas symmetry conditions are applied on the boundary surface with trace AB, see Figure 15. The evolution with time of the proportional load coefficient p(t) amplifies the imposed displacement 6 according to
Figure 14. Notched specimen subjected to cyclic loading: cyclic response for purely kinematic hardening (solid line) and for purely isotropic hardening (dashed line). A circumferentially notched bar of circular cross section under axial loading cycles is considered [46]. The notched specimen is a shaft whose net section diameter measures 25.4 units and the cir- cumferential notch has a radius of 12.7 units, see Figures 15 and 16 for geometry and mesh details. Kinematic hardening model is used along with the following material properties: E = 2.1 x 10°, v = 0.3, oyo = 250, Hiso = 0, Akin = 1.1 x 10°, and Hy; = 1.0 x 103. Owing to the sym- metry of the problem, only one-eighth of the billet is modeled. A prescribed displacement 6 = 0.2 unit is applied to the lateral boundaries on the boundary surface with trace CD, whereas symmetry conditions are applied on the boundary surface with trace AB, see Figure 15. The evolution with time of the proportional load coefficient p(t) amplifies the imposed displacement 6 according to
Figure 15. Notched shaft under tensile load: geometry and mesh.
Figure 15. Notched shaft under tensile load: geometry and mesh.
Figure 16. Notched shaft under tensile load: geometry and mesh.
Figure 16. Notched shaft under tensile load: geometry and mesh.
Figure 17. Notched specimen subjected to cyclic loading: cyclic response for purely kinematic hardening (solid line) and for purely isotropic hardening (dashed line).
Figure 17. Notched specimen subjected to cyclic loading: cyclic response for purely kinematic hardening (solid line) and for purely isotropic hardening (dashed line).
In the following, we will indicate, respectively, the midpoint and endpoint values of the yield function (viz. Equation (42)) by F,4_ and F,41. Applying Dini’s theorem, it is found that
In the following, we will indicate, respectively, the midpoint and endpoint values of the yield function (viz. Equation (42)) by F,4_ and F,41. Applying Dini’s theorem, it is found that
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