Thermomechanical Characteristics of a Functionally Graded Mounted Uniform Disc with/without Rigid Casing

Yildirim, Vebil

doi:10.5028/jatm.v11.1008

Vebil Yildirim ^**Correspondence author: vebil@cu.edu.tr

.University of Çukurova - Engineering Faculty - Department of Mechanical Engineering - Adana - Turkey.

http://orcid.org/0000-0001-9955-8423

*Correspondence author: vebil@cu.edu.tr

Section Editor: Marcia Mantelli

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Figure 1
Mounted discs (a) without rigid casing (fi xed-free), and (b) with rigid casing (fixed-fixed).

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Figure 2
Variation of the dimensionless elastic fields with the inhomogeneity index under individual and combined loads for fixed-free boundary conditions.

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Figure 3
Variation of the dimensionless elastic fields with the inhomogeneity index under individual and combined loads for fixed-fixed boundary conditions.

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Figure 4
Variation of the radial displacements.

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Figure 5
Variation of the radial stresses.

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Figure 6
Variation of the hoop stresses.

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Table 1
Variation of the dimensionless elastic fields with the inhomogeneity index under individual and combined loads for mounted rotating disks.

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Table 2
Variation of the dimensionless equivalent stress under individual and combined loads for mounted rotating disks and β = 2, - 2.

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Table 3
Constituent material properties.

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Table 4
Elastic fields for physical FGMs under fixed-free boundary conditions.

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Table 5
Elastic fields for physical FGMs under fixed-fixed boundary conditions.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

	u_r /a		σ_r /pa		σ_θ /pa
r/a	β = -2	β = 2	β = -2	β = 2	β = -2	β = 2
Fixed-free
Thermal
1.00	0.00	0.00	0.00327254	0.00459112	-0.0470182	-0.0466227
1.04	0.0000005647	0.00000059	0.00166211	0.00276367	-0.0309148	-0.0389849
1.08	0.0000009674	0.00000105	0.00069278	0.00138161	-0.0186591	-0.0298611
1.12	0.0000012298	0.00000137	0.00017605	0.00045252	-0.00931328	-0.0191255
1.16	0.0000013704	0.00000155	-0.00002269	-0.00001265	-0.00218269	-0.00664791
1.2	0.0000014047	0.00000161	0.00	0.00	0.00325169	0.00770582
Centrifugal
1.00	0.00	0.00	0.300775	0.444601	0.0902324	0.13338
1.04	0.0000025014	0.00000354	0.232462	0.367487	0.0786334	0.124969
1.08	0.0000045049	0.00000614	0.168738	0.285286	0.064926	0.112092
1.12	0.0000059694	0.00000788	0.109049	0.197175	0.04971	0.0944611
1.16	0.0000068521	0.00000885	0.0529312	0.102346	0.0334387	0.0717666
1.2	0.0000071089	0.00000910	0.00	0.00	0.0164559	0.0436823
Combined (thermal + centrifugal)
1.	0.00	0.00	0.304047	0.449192	0.0432142	0.0867576
1.04	0.0000030660	0.00000413	0.234124	0.370251	0.0477187	0.0859837
1.08	0.0000054723	0.00000719	0.169431	0.286667	0.0462669	0.082231
1.12	0.0000071992	0.00000925	0.109225	0.197627	0.0403967	0.0753356
1.16	0.0000082224	0.00001040	0.0529085	0.102334	0.031256	0.0651187
1.2	0.0000085137	0.00001071	0.00	0.00	0.0197076	0.0513882
Fixed-fixed
Thermal
1.	0.00	0.00	-0.0245472	-0.0414957	-0.0553642	-0.0604487
1.04	0.0000003063	0.00000020	-0.0254226	-0.042107	-0.0399588	-0.0540915
1.08	0.0000004400	0.00000030	-0.0257345	-0.0424101	-0.0282619	-0.0462308
1.12	0.0000004220	0.00000031	-0.0256586	-0.0423822	-0.0193636	-0.0367431
1.16	0.0000002703	0.00000021	-0.0253195	-0.0419991	-0.0125908	-0.0255005
1.2	0.00	0.00	-0.0248054	-0.0412358	-0.00744163	-0.0123707
Centrifugal
1.	0.00	0.00	0.159987	0.183347	0.047996	0.0550041
1.04	0.0000011941	0.00000130	0.0953936	0.113127	0.0328641	0.0393328
1.08	0.0000018358	0.00000189	0.0349973	0.0370416	0.0163286	0.0192969
1.12	0.0000018813	0.00000185	-0.0216937	-0.0456441	-0.00115191	-0.00540828
1.16	0.0000012849	0.00000121	-0.0750889	-0.135664	-0.019234	-0.0351038
1.2	0.00	0.00	-0.125533	-0.233755	-0.03766	-0.0701265
Combined (thermal + centrifugal)
1.	0.00	0.00	0.135439	0.141851	-0.0073682	-0.00544463
1.04	0.0000015004	0.00000150	0.069971	0.0710199	-0.00709467	-0.0147588
1.08	0.0000022758	0.00000220	0.00926279	-0.00536849	-0.0119334	-0.0269339
1.12	0.0000023033	0.00000216	-0.0473523	-0.0880263	-0.0205155	-0.0421513
1.16	0.0000015552	0.00000141	-0.100408	-0.177663	-0.0318248	-0.0606043
1.2	0.00	0.00	-0.150339	-0.274991	-0.0451017	-0.0824972

σ_eq /pa
r/a	β = -2	β = 2	β = -2	β = 2
	Fixed-free		Fixed-fixed
1.	0.284909	0.41271	0.13927	0.14465
1.04	0.214287	0.335624	0.0737747	0.0794344
1.08	0.151685	0.25567	0.0184049	0.0246914
1.12	0.0956535	0.172753	0.0411299	0.0762557
1.16	0.046071	0.089715	0.0888773	0.156429
1.2	0.0197076	0.0513882	0.133624	0.244417

Materials		E (GPa)	ρ (kg/m³)	ν	k (W/mK)	α (1/K)
Metals	Nickel (Ni)	199.5	8900	0.3	90.7	13.3 10^-6
	Aluminum (Al)	70	2700	0.3	204	23. 10^-6
	SUS304 (Stainless Steel)	201.04	7800	0.3262	15.379	12.33 10^-6
Ceramics	Silicon Nitride (Si3N4)	348.43	4429	0.24	1.209	5.8723 10^-6
	Aluminum Oxide (Al2O3)	393	3970	0.3	30.1	8.8 10^-6
	Zirconium oxide (ZrO2)	116.4	3657	0.3	1.78	8.7 10^-6

r/b	Thermal			Centrifugal			Thermal + centrifugal
r/b	S_i3N₄	Ni-S_i3N₄	Ni	S_i3N₄	Ni-S_i3N₄	Ni	S_i3N₄	Ni-S_i3N₄	S_i3N₄
u_r (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.3227	0.3542	0.74496	0.0046	0.008080	0.0154	0.3273	0.3623	0.7604
0.7	0.5871	0.6918	1.35457	0.0080	0.015174	0.0266	0.5951	0.7070	1.3811
0.8	0.8132	1.0405	1.87476	0.0103	0.020926	0.0341	0.8234	1.0614	1.9088
0.9	1.0121	1.4112	2.33162	0.0115	0.024729	0.0381	1.0236	1.4359	2.3697
1.0	1.1908	1.8090	2.74121	0.0118	0.025754	0.0387	1.2026	1.8347	2.7799
σ_r (GPa)
0.5	0.3346	0.3895	0.42067	0.0199	0.032058	0.0393	0.3545	0.4215	0.4600
0.6	0.1835	0.2383	0.230133	0.0153	0.025922	0.0305	0.1988	0.2642	0.2606
0.7	0.0989	0.1495	0.123707	0.0114	0.020327	0.0228	0.1103	0.1698	0.1465
0.8	0.0490	0.0874	0.061041	0.0077	0.014430	0.0154	0.0567	0.1019	0.0765
0.9	0.0186	0.0395	0.023099	0.0039	0.007758	0.0079	0.0225	0.0473	0.0310
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σ_θ (GPa)
0.5	-0.683	-0.658	-0.86350	0.0048	0.008656	0.0118	-0.678	-0.649	-0.852
0.6	-0.478	-0.425	-0.60317	0.0064	0.011051	0.0143	-0.472	-0.413	-0.589
0.7	-0.348	-0.357	-0.43774	0.0067	0.011250	0.0144	-0.341	-0.345	-0.423
0.8	-0.259	-0.342	-0.32395	0.0063	0.010141	0.0131	-0.252	-0.332	-0.311
0.9	-0.193	-0.348	-0.24092	0.0054	0.008061	0.0108	-0.188	-0.340	-0.230
1.0	-0.144	-0.363	-0.17749	0.0041	0.005138	0.0077	-0.140	-0.358	-0.170
r/b	Al₂O₃	Al-Al₂O₃	Al	Al₂O₃	Al-Al₂O₃	Al	Al₂O₃	Al-Al₂O₃	Al
u_r (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.4929	0.5326	1.28828	0.0035	0.003615	0.0133	0.4964	0.5362	1.3016
0.7	0.8963	1.0872	2.34249	0.0060	0.007549	0.0230	0.9023	1.0947	2.3655
0.8	1.2404	1.6815	3.24206	0.0077	0.011338	0.0295	1.2482	1.6928	3.2715
0.9	1.5427	2.3249	4.03212	0.0086	0.014237	0.0330	1.5514	2.3392	4.0651
1.0	1.8137	3.0226	4.74044	0.0088	0.015190	0.0335	1.8225	3.0377	4.7739
σ_r (GPa)
0.5	0.5483	0.4507	0.255254	0.0175	0.014537	0.0119	0.5658	0.4653	0.2672
0.6	0.3000	0.2337	0.13964	0.0136	0.011001	0.0092	0.3136	0.2447	0.1489
0.7	0.1612	0.1273	0.075063	0.0102	0.008010	0.0069	0.1714	0.1353	0.0820
0.8	0.0796	0.0662	0.037039	0.0069	0.005253	0.0047	0.0864	0.0714	0.0417
0.9	0.0301	0.0270	0.014016	0.0035	0.002604	0.0024	0.0336	0.0296	0.0164
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σ_θ (GPa)
0.5	-1.125	-1.155	-0.52395	0.0053	0.004361	0.0036	-1.120	-1.150	-0.520
0.6	-0.786	-0.632	-0.36599	0.0064	0.004804	0.0043	-0.780	-0.627	-0.362
0.7	-0.571	-0.418	-0.26561	0.0064	0.004237	0.0044	-0.564	-0.414	-0.261
0.8	-0.422	-0.316	-0.19657	0.0059	0.003305	0.0040	-0.416	-0.313	-0.193
0.9	-0.314	-0.261	-0.14619	0.0048	0.002221	0.0033	-0.309	-0.259	-0.143
1.0	-0.231	-0.228	-0.10770	0.0034	0.001063	0.0023	-0.228	-0.227	-0.105
r/b	ZrO₂	SUS304-ZrO₂	SUS304	ZrO₂	SUS304-ZrO₂	SUS304	ZrO₂	SUS304-ZrO₂	SUS304
u_r (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.4873	0.5370	0.696216	0.0109	0.016374	0.0131	0.4982	0.5533	0.7093
0.7	0.8861	0.9831	1.2655	0.0187	0.027204	0.0225	0.9048	1.0104	1.2880
0.8	1.2263	1.3864	1.7509	0.0240	0.033985	0.0288	1.2503	1.4204	1.7797
0.9	1.5252	1.7698	2.17686	0.0268	0.037395	0.0322	1.5520	1.8072	2.2090
1.0	1.7931	2.1458	2.55842	0.0273	0.037721	0.0326	1.8204	2.1835	2.5910
σ_r (GPa)
0.5	0.1606	0.2378	0.387834	0.0162	0.025836	0.0342	0.1767	0.2636	0.4220
0.6	0.0878	0.1529	0.211934	0.0125	0.021015	0.0266	0.1004	0.1739	0.2385
0.7	0.0472	0.0976	0.113778	0.0094	0.016592	0.0200	0.0566	0.1142	0.1337
0.8	0.0233	0.0577	0.056058	0.0063	0.011878	0.0135	0.0296	0.0696	0.0696
0.9	0.0088	0.0263	0.021176	0.0032	0.006448	0.0069	0.0121	0.0328	0.0281
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σ_θ (GPa)
0.5	-0.330	-0.303	-0.79809	0.0048	0.008089	0.0112	-0.325	-0.295	-0.787
0.6	-0.230	-0.247	-0.55699	0.0059	0.010247	0.0131	-0.224	-0.237	-0.544
0.7	-0.167	-0.225	-0.40371	0.0059	0.011093	0.0130	-0.161	-0.214	-0.391
0.8	-0.124	-0.221	-0.29823	0.0054	0.010882	0.0117	-0.118	-0.211	-0.287
0.9	-0.092	-0.230	-0.22123	0.0044	0.009706	0.0095	-0.088	-0.220	-0.212
1.0	-0.068	-0.245	-0.16237	0.0032	0.007583	0.0066	-0.065	-0.238	-0.156

r/b	Thermal			Centrifugal			Thermal + centrifugal
r/b	Si₃N₄	Ni-Si₃N₄	Ni	Si₃N₄	Ni-Si₃N₄	Ni	Si₃N₄	Ni-Si₃N₄	Si₃N₄
u_r (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.0315	0.0038	0.074886	0.0018	0.003091	0.0060	0.0333	0.0069	0.0808
0.7	0.0427	-0.009	0.101444	0.0026	0.005198	0.0089	0.0453	-0.004	0.1103
0.8	0.0392	-0.018	0.092971	0.0026	0.005859	0.0089	0.0418	-0.012	0.1019
0.9	0.0242	-0.016	0.057431	0.0018	0.004413	0.0060	0.0260	-0.011	0.0634
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σ_r (GPa)
0.5	-0.839	-0.943	-1.18188	0.0083	0.013091	0.0167	-0.831	-0.930	-1.165
0.6	-0.854	-0.945	-1.20104	0.0050	0.009080	0.0103	-0.849	-0.936	-1.191
0.7	-0.857	-0.940	-1.20412	0.0019	0.004812	0.0041	-0.855	-0.936	-1.200
0.8	-0.853	-0.938	-1.19972	-0.001	-0.00018	-0.002	-0.854	-0.939	-1.202
0.9	-0.847	-0.939	-1.19168	-0.005	-0.00618	-0.009	-0.852	-0.945	-1.201
1.0	-0.839	-0.942	-1.18188	-0.008	-0.01341	-0.017	-0.848	-0.956	-1.199
σ_θ (GPa)
0.5	-0.965	-1.018	-1.34426	0.0020	0.003535	0.0050	-0.963	-1.014	-1.339
0.6	-0.896	-0.920	-1.25532	0.0022	0.004002	0.0051	-0.894	-0.916	-1.250
0.7	-0.848	-0.917	-1.19322	0.0018	0.003273	0.0037	-0.846	-0.913	-1.189
0.8	-0.812	-0.935	-1.14651	0.0008	0.001701	0.0015	-0.811	-0.933	-1.145
0.9	-0.784	-0.957	-1.10947	-0.0004	-0.00060	-0.001	-0.784	-0.958	-1.111
1.0	-0.760	-0.979	-1.07893	-0.002	-0.00362	-0.005	-0.762	-0.982	-1.084
r/b	Al₂O₃	Al-Al₂O₃	Al	Al₂O₃	Al-Al₂O₃	Al	Al₂O₃	Al-Al₂O₃	Al
u_r (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.0495	0.1991	0.129503	0.0013	0.001939	0.0051	0.0509	0.2010	0.1347
0.7	0.0671	0.3086	0.175429	0.0020	0.003636	0.0077	0.0691	0.3122	0.1831
0.8	0.0615	0.3219	0.160776	0.0020	0.004505	0.0077	0.0635	0.3264	0.1685
0.9	0.0380	0.2250	0.099317	0.0014	0.003684	0.0052	0.0394	0.2287	0.1045
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σ_r (GPa)
0.5	-1.540	-0.786	-0.71714	0.0074	0.008323	0.0051	-1.533	-0.777	-0.712
0.6	-1.565	-0.868	-0.72877	0.0046	0.005462	0.0031	-1.561	-0.863	-0.726
0.7	-1.569	-0.887	-0.73064	0.0018	0.002914	0.0012	-1.568	-0.884	-0.729
0.8	-1.564	-0.883	-0.72797	-0.002	0.000484	-0.0007	-1.565	-0.882	-0.729
0.9	-1.553	-0.871	-0.72309	-0.004	-0.00191	-0.003	-1.557	-0.872	-0.726
1.0	-1.540	-0.855	-0.71714	-0.007	-0.00430	-0.005	-1.548	-0.860	-0.722
σ_θ (GPa)
0.5	-1.752	-1.526	-0.81567	0.0022	0.002500	0.0015	-1.750	-1.523	-0.814
0.6	-1.636	-1.101	-0.76170	0.0023	0.002445	0.0015	-1.634	-1.099	-0.760
0.7	-1.555	-0.912	-0.72403	0.0017	0.001758	0.0011	-1.554	-0.910	-0.723
0.8	-1.494	-0.808	-0.69568	0.0007	0.000832	0.0005	-1.494	-0.807	-0.695
0.9	-1.446	-0.743	-0.67320	-0.0006	-0.00020	-0.0004	-1.447	-0.743	-0.674
1.0	-1.406	-0.696	-0.65467	-0.002	-0.00129	-0.002	-1.409	-0.697	-0.656
r/b	ZrO₂	SUS304-ZrO₂	SUS304	ZrO₂	SUS304-ZrO₂	SUS304	ZrO₂	SUS304-ZrO₂	SUS304
u_r (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.0490	-0.109	0.070824	0.0042	0.005022	0.0051	0.0532	-0.104	0.0759
0.7	0.0664	-0.152	0.095941	0.0062	0.007250	0.0076	0.0726	-0.145	0.1035
0.8	0.0608	-0.142	0.087927	0.0063	0.007110	0.0076	0.0671	-0.135	0.0955
0.9	0.0376	-0.090	0.054315	0.0042	0.004710	0.0051	0.0418	-0.085	0.0594
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σ_r (GPa)
0.5	-0.451	-0.740	-1.14708	0.0069	0.008655	0.0146	-0.444	-0.731	-1.132
0.6	-0.458	-0.723	-1.16497	0.0042	0.005626	0.0091	-0.454	-0.717	-1.156
0.7	-0.460	-0.719	-1.16785	0.0017	0.002241	0.0036	-0.458	-0.717	-1.164
0.8	-0.458	-0.724	-1.16374	-0.001	-0.00186	-0.002	-0.459	-0.725	-1.166
0.9	-0.455	-0.734	-1.15623	-0.004	-0.00692	-0.008	-0.459	-0.741	-1.164
1.0	-0.451	-0.749	-1.14708	-0.007	-0.01316	-0.015	-0.458	-0.762	-1.162
σ_θ (GPa)
0.5	-0.513	-0.609	-1.29878	0.0021	0.002710	0.0048	-0.511	-0.607	-1.294
0.6	-0.479	-0.666	-1.21568	0.0021	0.002886	0.0047	-0.477	-0.663	-1.211
0.7	-0.455	-0.727	-1.15767	0.0015	0.002273	0.0034	-0.454	-0.724	-1.154
0.8	-0.438	-0.788	-1.11403	0.0006	0.000917	0.0013	-0.437	-0.787	-1.113
0.9	-0.423	-0.850	-1.07943	-0.0006	-0.00120	-0.001	-0.424	-0.851	-1.081
1.0	-0.412	-0.911	-1.0509	-0.002	-0.00412	-0.005	-0.414	-0.915	-1.056

\begin{matrix} σ_{r} (r) = C_{11} (r) ɛ_{r} + C_{12} (r) ɛ_{θ} - (C_{11} (r) + C_{12} (r)) α (r) T (r) \\ σ_{θ} (r) = C_{12} (r) ɛ_{r} + C_{11} (r) ɛ_{θ} - (C_{11} (r) + C_{12} (r)) α (r) T (r) \end{matrix}

\begin{matrix} u_{r} {(r)}^{″} + (\frac{1}{r} + \frac{{C^{'}}_{11} (r)}{C_{11} (r)}) {u^{'}}_{r} (r) + (- \frac{1}{r^{2}} + \frac{v}{r} \frac{{C^{'}}_{11} (r)}{C_{11} (r)}) u_{r} (r) \\ = - \frac{ρ (r) ω^{2} r}{C_{11} (r)} + (\frac{{C^{'}}_{11} (r)}{C_{11} (r)} α (r) + α^{'} (r)) (1 + v) T (r) + (1 + v) α (r) T^{'} (r) \end{matrix}

\frac{(- 1 + β v)}{r^{2}} u_{r} (r) + \frac{(1 + β)}{r} {u^{'}}_{r} (r) + {u^{″}}_{r} (r) =^{.} - \frac{a^{- q + β} r^{1 + q - β} (1 - v^{2}) ρ_{a} ω^{2}}{E_{a}} + r^{- 1 + n - µ} (r^{µ} ∆_{1} + ∆_{2})

\begin{matrix} u_{r} (r) = r^{- β} (r^{\frac{β - ξ}{2}} (B_{1} + B_{2} r^{ξ}) + r^{3 + q} Ω) \\ + ∆_{3} (r^{1 + n - µ} ∆_{4} + r^{1 + n} ∆_{1} ∆_{5} + C_{1} r^{\frac{1}{2} (- β - ξ)} ∆_{5} ∆_{6} + C_{2} r^{\frac{1}{2} (- β + ξ)} ∆_{5} ∆_{6}) \\ σ_{r} (r) = \frac{1}{2} r^{- 1 - β - \frac{ξ}{2}} C_{11} (r) (r {}^{β / 2}{(- B_{1} (β - 2 v + ξ) + B_{2} r^{ξ} (- β + 2 v + ξ))} \\ + 2 r^{3 + q + \frac{ξ}{2}} (3 + q - β + v) Ω) \\ + C_{11} (r) r^{n} ∆_{3} (∆_{1} ∆_{5} (1 + n + v) + r^{- µ} ∆_{4} (1 + n - µ + v)) \\ + \frac{1}{2} C_{11} (r) r^{\frac{1}{2} (- 2 - β - ξ)} ∆_{3} ∆_{5} ∆_{6} (- C_{1} (β - 2 v + ξ) + C_{2} r^{ξ} (- β + 2 v + ξ)) \\ + \frac{a^{- n - β} E_{a} r^{n + β - µ} α_{a} (r^{µ} Φ_{1} + Φ_{2})}{- 1 + v} \\ σ_{θ} (r) = \frac{1}{2} r^{- 1 - β - \frac{ξ}{2}} C_{11} (r) (- r^{β / 2} (- B_{2} r^{ξ} (2 - β v + v ξ) + B_{1} (- 2 + v (β + ξ))) \\ + 2 r^{3 + q + \frac{ξ}{2}} (1 + (3 + q - β) v) Ω) \\ + C_{11} (r) r^{n} ∆_{3} (∆_{1} ∆_{5} (1 + v + nv) + r^{- µ} (∆_{4} + ∆_{4} (1 + n - µ) v)) \\ + \frac{1}{2} C_{11} (r) r^{\frac{1}{2} (- 2 - β - ξ)} ∆_{3} ∆_{5} ∆_{6} (C_{2} r^{ξ} (2 - β v + v ξ) - C_{1} (- 2 + v (β + ξ))) \\ + \frac{a^{- n - β} E_{a} r^{n + β - µ} α_{a} (r^{µ} Φ_{1} + Φ_{2})}{- 1 + v} \end{matrix}

\begin{matrix} ξ = \sqrt{4 + β^{2} - 4 β v} \\ Ω = \frac{a^{- q + β} (- 1 + v^{2}) ρ_{a} ω^{2}}{E_{a} (8 + q (6 + q - β) - 3 β + β v)} \\ ∆_{3} \frac{16}{(- 2 - 2 n - β + ξ) (2 + 2 n + β + ξ) (2 + 2 n + β - 2 µ + ξ) (- 2 - 2 n - β + 2 µ + ξ)} \\ ∆_{4} = ∆_{2} (β + n (2 + n + β) + β v) \\ ∆_{5} = n^{2} + n (2 + β - 2 µ) + (- 2 + µ) µ + β (1 - µ + v) \\ ∆_{6} = β + n (2 + n + β) + β v \end{matrix}

\begin{matrix} B_{1} = \frac{Ω a^{\frac{ξ - β}{2}} b^{\frac{ξ - β}{2}} (a^{q + 3} b^{\frac{β + ξ}{2}} (β - 2 v - ξ) - 2 b^{q + 3} a^{\frac{β + ξ}{2}} (β - v - q - 3))}{a^{ξ} (β - 2 v + ξ) + b^{ξ} (- β + 2 v + ξ)} \\ B_{2} = - \frac{Ω a^{- β / 2} b^{- β / 2} (b^{β / 2} a^{\frac{ξ}{2} + q + 3} (β - 2 v + ξ) - 2 a^{β / 2} b^{\frac{ξ}{2} + q + 3} (β - v - q - 3))}{a^{ξ} (β - 2 v + ξ) + b^{ξ} (- β + 2 v + ξ)} \\ C_{1} = \frac{b^{\frac{1}{2} (ξ - 2 µ)} a^{\frac{1}{2} (ξ - 2 (µ + n))}}{∆_{3} ∆_{5} ∆_{6} (a^{ξ} (β - 2 v + ξ) + b^{ξ} (- β + 2 v + ξ))} (∆_{3} a^{\frac{β}{2} + 2 n + 1} b^{µ + \frac{ξ}{2}} (β - 2 v - ξ) (∆_{1} ∆_{5} a^{µ} + ∆_{4}) \\ + 2 a^{µ + \frac{ξ}{2}} b^{\frac{β}{2} + n + 1} (∆_{3} a^{n} (∆_{1} ∆_{5} b^{µ} (v + n + 1) + ∆_{4} (- µ + v + n + 1)) \\ - α_{a} (v + 1) (Φ_{1} b^{µ} + Φ_{2}))) \\ C_{2} = \frac{b^{- µ} a^{- µ - n}}{∆_{3} ∆_{5} ∆_{6} (a^{ξ} (β - 2 v + ξ) + b^{ξ} (- β + 2 v + ξ))} \\ (- ∆_{3} b^{µ} (β - 2 v + ξ) a^{\frac{1}{2} (β + 4 n + ξ + 2)} (∆_{1} ∆_{5} a^{µ} + ∆_{4}) \\ - 2 a^{µ} b^{\frac{1}{2} (β + 2 n + ξ + 2)} (∆_{3} a^{n} (∆_{1} ∆_{5} b^{µ} (v + n + 1) + ∆_{4} (- µ + v + n + 1)) \\ - α_{a} (v + 1) (Φ_{1} b^{µ} + Φ_{2}))) \end{matrix}

\begin{matrix} B_{1} = - \frac{Ω a^{ξ} b^{- \frac{β}{2} + \frac{ξ}{2} + q + 3} - Ω b^{ξ} a^{- \frac{β}{2} + \frac{ξ}{2} + q + 3}}{a^{ξ} - b^{ξ}} \\ B_{2} = \frac{Ω (b^{- \frac{β}{2} + \frac{ξ}{2} + q + 3} - a^{- \frac{β}{2} + \frac{ξ}{2} + q + 3})}{a^{ξ} - b^{ξ}} \\ C_{1} = - \frac{a^{ξ} (∆_{1} ∆_{5} b^{µ} + ∆_{4}) b^{\frac{1}{2} (β - 2 µ + 2 n + ξ + 2)} - b^{ξ} (∆_{1} ∆_{5} a^{µ} + ∆_{4}) a^{\frac{1}{2} (β - 2 µ + 2 n + ξ + 2)}}{∆_{5} ∆_{6} (a^{ξ} - b^{ξ})} \\ C_{2} = - \frac{(∆_{1} ∆_{5} a^{µ} + ∆_{4}) a^{\frac{1}{2} (β - 2 µ + 2 n + ξ + 2)} - (∆_{1} ∆_{5} b^{µ} + ∆_{4}) b^{\frac{1}{2} (β - 2 µ + 2 n + ξ + 2)}}{∆_{5} ∆_{6} (a^{ξ} - b^{ξ})} \end{matrix}

\begin{matrix} u_{r} (r) = \frac{(v^{2} - 1) {ρ ω}^{2} (a - r) (a + r) (a^{2} (b^{2} (v + 1) - (v - 1) r^{2}) + b^{2} ((v + 1) r^{2} - b^{2} (v + 3)))}{8 Er (a^{2} (v - 1) - b^{2} (v + 1))} \\ + \frac{1}{2 a^{2} (v - 1) r - 2 b^{2} (v + 1) r} \{a^{2} κ \ln a (b^{2} (v + 1) - (v - 1) r^{2}) - κ r^{2} \ln r (b^{2} (v + 1) - a^{2} (v - 1)) \\ - b^{2} (a - r) (a + r) (- 2 α (v + 1) Ψ_{1} + (v + 1) \ln b (κ - 2 α Ψ_{2}) + κ)\} \\ σ_{r} (r) = \frac{{ρ ω}^{2} (r^{2} - b^{2}) (a^{4} (v^{2} - 1) - a^{2} (v - 1) (v + 3) (b^{2} + r^{2}) + b^{2} (v + 1) (v + 3) r^{2})}{8 r^{2} (a^{2} (v - 1) - b^{2} (v + 1))} \\ + \frac{1}{2 (v^{2} - 1) r^{2} (a^{2} (v - 1) - b^{2} (v + 1))} \{E ((v + 1) (κ - 2 α Ψ_{2}) (b^{2} \ln b (a^{2} (v - 1) - (v + 1) r^{2}) + \\ + r^{2} \ln r (b^{2} (v + 1) - a^{2} (v - 1))) + a^{2} (v - 1) (b - r) (b + r) (κ - 2 α (v + 1) Ψ_{1}) - \\ - a^{2} κ (v^{2} - 1) \ln a (b - r) (b + r))\} \\ σ_{θ} (r) = \frac{1}{8 r^{2} (a^{2} (v - 1) - b^{2} (v + 1))} ({ρ ω}^{2} (a^{4} (v^{2} - 1) (b^{2} + r^{2}) - a^{2} (v - 1) (b^{4} (v + 3) + \\ + (3 v + 1) r^{4}) - b^{2} (v + 1) r^{2} (b^{2} (v + 3) - (3 v + 1) r^{2}))) \\ + \frac{1}{2 (v^{2} - 1) r^{2} (a^{2} (v - 1) - b^{2} (v + 1))} \{E (a^{2} (v - 1) (2 α (v + 1) Ψ_{1} (b {}^{2}+ r^{2}) - κ (b^{2} + {vr}^{2})) \\ - b^{2} \ln b (a^{2} (v - 1) + (v + 1) r^{2})) + a^{2} κ (v^{2} - 1) \ln a (b^{2} + r^{2}) + b^{2} κ (v^{2} - 1) r^{2})\} \end{matrix}

\begin{matrix} u_{r} (r) = \frac{(v^{2} - 1) {ρ ω}^{2} (r^{2} - a^{2}) (r^{2} - b^{2})}{8 Er} \\ + \frac{κ (b^{2} (r^{2} - a^{2}) \ln b + a^{2} \ln a (b - r) (b + r) + r^{2} (a - b) (a + b) \ln r)}{2 r (a - b) (a + b)} \\ σ_{r} (r) = \frac{{ρ ω}^{2} (a^{2} ((v + 1) r^{2} - b^{2} (v - 1)) + r^{2} (b^{2} (v + 1) - (v + 3) r^{2}))}{8 r^{2}} \\ + \frac{E}{2 (v - 1) (v + 1) r^{2} (a - b) (a + b)} \{(a^{2} κ \ln a ((v + 1) r^{2} - b^{2} (v - 1)) \\ + b^{2} κ \ln b (a^{2} (v - 1) - (v + 1) r^{2}) \\ - r^{2} (a - b) (a + b) (- 2 α (v + 1) Ψ_{1} + κ + (v + 1) \ln r (κ - 2 α Ψ_{2})))\} \\ σ_{θ} (r) = \frac{{ρ ω}^{2} (a^{2} (b^{2} (v - 1) + (v + 1) r^{2}) + r^{2} (b^{2} (v + 1) - (3 v + 1) r^{2}))}{8 r^{2}} \\ + \frac{E}{2 (v - 1) (v + 1) r^{2} (a - b) (a + b)} \{(a^{2} κ \ln a (b^{2} (v - 1) + (v + 1) r^{2}) \\ - b^{2} κ \ln b (a^{2} (v - 1) + (v + 1) r^{2}) \\ - r^{2} (a - b) (a + b) (- 2 α (v + 1) Ψ_{1} + κ v + (v + 1) \ln r (κ - 2 α Ψ_{2})))\} \end{matrix}

\begin{matrix} a = 1.0 m; b = 1.2 m; ω = 100 rad / s \\ v = 0.3; E_{a} = 200 GPa; ρ_{a} = 7800 \frac{kg}{m^{3}} \\ α_{a} = 1.2 10^{- 6} \frac{1}{° C}; k_{a} = 15.379 \frac{W}{m ° C} \\ T_{a} = 10 ° C; T_{b} = 0 ° C \end{matrix}

[1] *Correspondence author: vebil@cu.edu.tr

Brasil

Brasil

Thermomechanical Characteristics of a Functionally Graded Mounted Uniform Disc with/without Rigid Casing

ABSTRACT