Test 4: Cracks at a hole in a plate

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Remarks
Input files

ProductsAbaqus/Standard

Elements tested

CPE8

CPE8R

CPS8

CPS8R

Problem description

Mesh:

Collapsed elements with 1/4 point midside nodes are used at the crack tip. One-quarter of the test geometry is modeled.

Material:

Young's modulus = 207 GPa, Poisson's ratio = 0.3

Boundary conditions:

$u_{x} = 0$ along edge AB, $u_{y} = 0$ along edge ED.

Loading:

Uniform stress, $σ_{y y}$ = 100 N/mm².

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 4.1 and 4.2 from NAFEMS publication “2D Test Cases in Linear Elastic Fracture Mechanics,” R0020.

Target solution: K $I$ /K $_{0}$ = 1.050, K $_{0}$ = $σ \sqrt{π a}$

Results and discussion

The results are shown in the following table. The values enclosed in parentheses are percentage differences with respect to the reference solution.


Element Type	K $I$ /K $_{0}$
CPE8	1.060 (+0.9%)
CPE8R	1.059 (+0.8%)
CPS8	1.060 (+1.0%)
CPS8R	1.059 (+0.8%)

Remarks

K $I$ = $\sqrt{J E^{'}}$ . $E^{'} = E$ for plane stress, $E^{'} = E / (1 - ν^{2})$ for plane strain. An average of the J values calculated by Abaqus, excluding the first contour, is used in reporting the results. Experience has shown that the crack-tip elements do not give sufficiently accurate results to give good estimates of the J-integral for the first contour.

Input files

nlf41f8x.inp: CPE8 elements.
nlf41r8x.inp: CPE8R elements.
nlf42f8x.inp: CPS8 elements.
nlf42r8x.inp: CPS8R elements.