Test 3: Angle crack embedded 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

Problem description

Mesh:

Collapsed elements with 1/4 point midside nodes are used at the crack tip. The complete test geometry is modeled.

Material:

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

Boundary conditions:

$u_{y} = 0$ along edge AD, $u_{x} = 0$ at point D.

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 3.1 and 3.2 from NAFEMS publication “2D Test Cases in Linear Elastic Fracture Mechanics,” R0020.

Target solution ( $β$ = 22.5°): K $I$ /K $_{0}$ = 0.190, K $_{I I}$ /K $_{0}$ = 0.405, K $_{0}$ = $σ \sqrt{π a}$

Target solution ( $β$ = 67.5°): K $I$ /K $_{0}$ = 1.030, K $_{I I}$ /K $_{0}$ = 0.370, 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.


$β$	Element Type	K $I$ /K $_{0}$	K $_{I I}$ /K $_{0}$
22.5°	CPE8	0.185 (−2.9%)	0.405 (+0.1%)
22.5°	CPE8R	0.184 (−2.9%)	0.407 (+0.4%)
67.5°	CPE8	1.035 (+0.2%)	0.364 (−1.7%)
67.5°	CPE8R	1.038 (+0.8%)	0.368 (−0.5%)

Remarks

K $I$ = $\sqrt{\frac{J E}{(1 + R^{2}) (1 - ν^{2})}}$ , K $_{I I}$ = RK $I$ , $R = \frac{t \cdot (u^{+} - u^{-})}{n \cdot (u^{+} - u^{-})}$ .

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. $u^{+}$ and $u^{-}$ are the displacements of nodes on the positive and negative sides of the crack, respectively, that are initially located at the same position in the undeformed state. An average value for R based on the first five nodal locations behind (not including) the crack tip was used in the calculations. $t$ and $n$ are the tangent and normal, respectively, to the direction of crack propagation.

Input files

nlf31f8x.inp: CPE8 elements, Test 3.1.
nlf31r8x.inp: CPE8R elements, Test 3.1.
nlf32f8x.inp: CPE8 elements, Test 3.2.
nlf32r8x.inp: CPE8R elements, Test 3.2.