Test 1.2: Center cracked plate with thermal load

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

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, thermal expansion coefficent = 1.35E−5.

Boundary conditions:

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

Loading:

Quadratic thermal distribution T = T₀ $\frac{x^{2}}{c^{2}}$ = 0.01x², where T₀ = 100, c = 100.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 1.2 from NAFEMS publication “2D Test Cases in Linear Elastic Fracture Mechanics,” R0020.

Target solution: K $I$ /K $_{0}$ = 1.000, K $_{0}$ = $\frac{E α T}{c^{2}} (\frac{b^{2}}{3} - \frac{a^{2}}{2}) \sqrt{π a}$

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 $_{0}$
CPS8	1.003 (+0.3%)
CPS8R	1.005 (+0.5%)

Remarks

K $I$ = $\sqrt{J E}$ . 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. The thermal loading is applied with user subroutine UTEMP.

Input files

nlf12f8x.inp: CPS8 elements.
nlf12f8x.f: User subroutine UTEMP used in nlf12f8x.inp.
nlf12r8x.inp: CPS8R elements.
nlf12r8x.f: User subroutine UTEMP used in nlf12r8x.inp.