Finite element formulations of strain gradient theory for microstructures and the <i>C</i><sup>0–1</sup> patch test

Soh, Ai‐Kah; Wanji, Chen

doi:10.1002/nme.1075

Cited by 63 publications

(40 citation statements)

References 22 publications

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“…However, a strict C 1 element is complicated to construct and implement. Alternatively, Soh and Chen [26] advocated using two types of displacement functions with same nodal parameters: one satisfies C 0 continuity and quadratic completeness to calculate first derivatives of displacement, and the other satisfies C 1 weak continuity to calculate second derivatives of displacement. This approach has been used to develop an effective 18-DOF plane couple stress/strain gradient triangular element [26].…”

Section: A Weak Continuity Condition Of Axisymmetric Element For Coupmentioning

confidence: 99%

See 1 more Smart Citation

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

Zhao¹,

Wanji²,

Ji³

2010

Finite Elements in Analysis and Design

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Section: A Weak Continuity Condition Of Axisymmetric Element For Coupmentioning

confidence: 99%

“…Soh and Chen [26] established the C 0 À 1 patch test for the finite element analysis of couple stress/strain gradient theory. It can be summarized as follows.…”

Section: Numerical Examplesmentioning

confidence: 99%

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

Zhao¹,

Wanji²,

Ji³

2010

Finite Elements in Analysis and Design

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“…In other words, the displacement function of the patch test has to be a complete quadratic function which satisfies the equilibrium equation. In the papers [15,16] , the authors proposed the enhanced C 0-1 patch test condition for the couple stress/strain gradient theory and established an 18-parameter triangular element that satisfies the C 0-1 patch test condition.…”

Section: -1 Patch Test For the Couple Stress/strain Gradient Theorymentioning

confidence: 99%

“…Therefore, the above mentioned obstacles in the patch test still exist. In the papers [15,16] , further discussion of the patch test found that the element for couple stress/strain gradient problem was not able to pass the constant curvature C 1 patch test. Because the test function which described constant strain curvature of the strain gradient theory cannot pass the C 0 patch test in state of the constant membrane stresses, the C 0-1 patch test is the test of linear membrane stress and constant strain curvature for the strain gradient theory/couple stress theory, but not both of constant stresses.…”

Section: Introductionmentioning

confidence: 98%

Enhanced patch test of finite element methods

Wanji

2006

SCI CHINA SER G

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Theoretically, the constant stress patch test is not rigorous. Also, either the patch test of non-zero constant shear for Mindlin plate problem or non-zero strain gradient curvature of the microstructures cannot be performed. To improve the theory of the patch test, in this paper, based on the variational principle with relaxed continuity requirement of nonconforming element for homogeneous differential equations, the author proposed the individual element condition for passing the patch test and the convergence condition of the element: besides passing the patch test, the element function should include the rigid body modes and constant strain modes and satisfy the weak continuity condition, and no extra zero energy modes occur. Moreover, the author further established a variational principle with relaxed continuity requirement of nonconforming element for inhomogeneous differential equations, the enhanced patch test condition and the individual element condition. To assure the convergence of the element that should pass the enhanced patch test, the element function should include the rigid body modes and non-zero strain modes which satisfied the equilibrium equations, and no spurious zero energy modes occur and should satisfy new weak continuity condition. The theory of the enhanced patch test proposed in this paper can be applied to both homogeneous and inhomogeneous differential equations. Based on this theory, the patch test of the non-zero constant shear stress for Mindlin plate and the C 0-1 patch test of the non-zero constant curvature for the couple stress/strain gradient theory were established.

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“…[6,7], the authors further studied the patch test for micro couple stress/strain gradient element. They proposed the enhanced C 0−1 patch test and established C 0−1 triangular elements with 18 parameters.…”

mentioning

confidence: 99%

Patch test function for axisymmetric element of conventional and couple stress theory

Wanji

Zhao

Wang

et al. 2009

Sci. China Ser. G-Phys. Mech. Astron.

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The enhanced patch test proposed by Chen W J (2006) can be used to assess the convergence of the problem with non-homogeneous differential equations. Based on this theory, we establish the patch test function for axisymmetric elements of conventional and couple stress theories, and reach an important conclusion that the patch test function for axisymmetric elements cannot contain non-zero constant shear.axisymmetric element, non-homogeneous differential equations, couple stress theory, test function of patch testIn finite element analysis, patch test has been used as criteria for assessing the convergence of the finite elements. The theory of patch test has been well developed [1] . However, the applications of the existing patch test are limited in the 2D/3D elasticity and thin plate problem with homogeneous differential equations. In addition, the phenomenon of zero shear stress in the patch test for conventional axisymmetric elements needs to be addressed. There is also need for establishing the patch test for micro couple stress/strain gradient finite element.In order to use patch test to assess the convergence of the element, the first step is to determine the patch test function. The patch test function for constant stress element is straightforward. For example, for 2D elasticity, u=a 1 +a 2 x+a 3 y, v=b 1 +b 2 x+b 3 y; for 3D elasticity, u=a 1 + a 2 x+a 3 y+a 4 z, v=b 1 +b 2 x+b 3 y+b 4 z, w=c 1 +c 2 x+c 3 y+c 4 z; for thin plate problem, 2 1 2 3 4 5 w a a x a y a x a xy = + + + + + 2 6 , a y where , , ( 1 4) i i i a b c i = … are arbitrary constants.The above patch test functions related to constant stress state satisfy the equilibrium equations respectively. In the finite element research of the axisymmetric element, lots of work are involved in how to do the patch test. However, majority of these previously proposed patch tests are performed by applying simple external forces. There has been no comprehensive and through study from the patch test function perspective. Currently, the patch test function for C 1 continuity couple stress axisymmetric element does not exist. The usage of linear displacement functions as patch test function for axisymmetric problem does not guarantee the satisfaction of patch test even for conforming elements. To improve this situation, Chen and Cheung [2] proposed new patch test function for axisymmetric problems. Based on weighted Sobolev spaces, Gao and Chen further established the convergence criterion for axisymmetric nonconforming elements, as well as generalized patch test and the function of the patch test [3,4] . The governing equations for axisymmetric problems in cylindrical coordinate are nonhomogeneous differential equations. The enhanced patch test function proposed in ref.[5] can be used in nonhomogeneous differential equations. The enhanced patch test method utilizes the dis-

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Finite element formulations of strain gradient theory for microstructures and the C^0–1 patch test

Cited by 63 publications

References 22 publications

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

Enhanced patch test of finite element methods

Patch test function for axisymmetric element of conventional and couple stress theory

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Finite element formulations of strain gradient theory for microstructures and the C0–1 patch test

Cited by 63 publications

References 22 publications

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular axisymmetric element

Enhanced patch test of finite element methods

Patch test function for axisymmetric element of conventional and couple stress theory

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Finite element formulations of strain gradient theory for microstructures and the C^0–1 patch test