Acoustic properties and low strain rate behavior of different types of chocolate

Figures & data

Figure 1. The dependence of the true strain rate on the engineering strain for white chocolate.

Table 1. Acoustical properties of the tested chocolates.

Chocolate	ρ (kg/m^3)	cL (m/s)	cT (m/s)
DC	1269.4	2208	798
MC	1263.4	2106	700
WC	1252.8	2094	649
RC	1244.8	2074	611
EDC	1170.0	1982	689

Table 2. Elastic properties of the tested chocolates.

Chocolate	E (MPa)	ν	G (MPa)	K (MPa)
DC	2303.6	0.4243	803.36	3110.8
MC	1780.3	0.4373	619.07	4778.1
WC	1712.0	0.4393	594.73	4700.3
RC	1516.0	0.4457	524.31	4655.4
EDC	1264.5	0.4475	436.79	4013.8

Figure 2. Example of the stress-strain curve for white chocolate. The difference between engineering and true stress strain dependences is shown in the upper part of the figure. The lower part shows the loading history of the specimen in the space strain-strain rate – stress.

Figure 3. (a.) Stress-strain curves at loading rate 1mm/min, (b.) Stress-strain curves at loading rate 10mm/min, (c.) Stress-strain curves at loading rate 100mm/min.

Figure 4. Effect of loading rate on the extent of the first region of the stress-strain curve.

Table 3. Parameters of linear function describing the engineering stress vs engineering strain dependence in the first region.

Chocolate	v (mm/min)	a (MPa)	b (MPa)	R^2	ε1 (1)
WC	1	−0.0348 ± 0.0004	13.2 ± 0.2	0.9964	0.023 ± 0.001
	10	−0.0298 ± 0.0002	12.9 ± 0.2	0.9964	0.0443 ± 0.0003
	100	−0.00077 ± 0.00005	10.8 ± 0.5	0.9989	0.0571 ± 0.0002
RC	1	−0.0355 ± 0.0007	8.1 ± 0.2	0.9917	0.05 ± 0.07
	10	−0.0905 ± 0.0008	11.6 ± 0.3	0.9812	0.0671 ± 0.0009
	100	0.00037 ± 0.00001	15.0 ± 0.2	0.9913	0.0857 ± 0.0008
MC	1	−0.100 ± 0.001	11.2 ± 0.2	0.9697	0.059 ± 0.001
	10	−0.127 ± 0.003	14.4 ± 0.3	0.9809	0.076 ± 0.002
	100	−0.181 ± 0.006	15.2 ± 0.4	0.9807	0.114 ± 0.007
EDC	1	0.024 ± 0.003	17.1 ± 0.2	0.9932	0.039 ± 0.002
	10	0.036 ± 0.007	22.3 ± 0.4	0.9964	0.0443 ± 0.0003
	100	−0.017 ± 0.001	20.2 ± 0.5	0.9985	0.0814 ± 0.0002
DC	1	−0.121 ± 0.005	14.8 ± 0.4	0.9750	0.059 ± 0.002
	10	−0.183 ± 0.004	28.2 ± 0.4	0.9932	0.076 ± 0.002
	100	−0.074 ± 0.004	24.7 ± 0.5	0.9304	0.114 ± 0.004

Table 4. Parameters of linear function describing the true stress vs true strain dependence in the first region.

Chocolate	v (mm/min)	a (MPa)	b (MPa)	R^2	ε1 (1)
WC	1	0.0258 ± 0.0004	12.5 ± 0.6	0.9901	0.025 ± 0.002
	10	−0.0277 ± 0.0002	12.4 ± 0.4	0.9967	0.045 ± 0.001
	100	0.016 ± 0.002	9.5 ± 0.4	0.9973	0.092 ± 0.001
RC	1	−0.0303 ± 0.0007	7.5 ± 0.4	0.9952	0.053 ± 0.002
	10	−0.0802 ± 0.0008	10.7 ± 0.8	0.9883	0.060 ± 0.001
	100	−0.0048 ± 0.0001	14.3 ± 0.6	0.9965	0.074 ± 0.001
MC	1	−0.088 ± 0.001	10.0 ± 0.2	0.9670	0.056 ± 0.004
	10	−0.105 ± 0.003	12.7 ± 0.5	0.9755	0.0664 ± 0.0007
	100	−0.144500 ± 0.0007	13.2 ± 0.2	0.9734	0.10 ± 0.06
EDC	1	0.039 ± 0.003	15.2 ± 0.1	0.9932	0.0364 ± 0.0007
	10	0.053 ± 0.003	20.4 ± 0.5	0.9947	0.043 ± 0.001
	100	0.0125 ± 0.0003	17.8 ± 0.3	0.9958	0.6950 ± 0.0005
DC	1	−0.0943 ± 0.0008	13.0 ± 0.3	0.9764	0.604 ± 0.001
	10	−0.1437 ± 0.0003	25.1 ± 0.3	0.9941	0.074 ± 0.004
	100	−0.073 ± 0.003	23.0 ± 0.4	0.9937	0.085 ± 0.002

Figure 5. Parameters of non linear function describing the dependence true stress vs true strain in the second region.

Figure 6. Parameters of linear function describing the engineering stress vs engineering strain dependence in the second region.

Figure 7. (a.) Effect of the loading rate on the stress vs strain dependence – engineering approach, (b.) Effect of the loading rate on the stress vs strain dependence – true values of stress and strain.

Figure 8. Example of linear fit of engineering stress vs engineering strain.

Table 5. Best fittings parameters of Equation (13)(13) ${\sigma }_{true}=a_4{\epsilon }_{true}^4+a_3{\epsilon }_{true}^3+a_2{\epsilon }_{true}^2+a_1{\epsilon }_{true}+a_0,$(13) and determination coefficients (R²) for true stress – true strain.

Chocolate	v (mm/min)	a4 (MPa)	a3 (MPa)	a2 (MPa)	a1 (MPa)	a0 (MPa)	R^2
WC	1	0	3.529	−3.889	1.403	0.2789	0.9957
	10	0	7.050	−8.540	2.641	0.3636	0.9888
	100	−104.400	146.100	−74.690	15.760	−0.0678	0.9973
RC	1	−20.852	28.323	−13.881	2.895	0.2373	0.9972
	10	−5.436	14.491	−11.342	3.167	0.4447	0.9769
	100	−25.362	36.599	−20.717	4.686	0.8053	0.9755
MC	1	−17.608	29.073	−17.573	4.506	0.3066	0.9972
	10	22.697	−12.099	−6.136	3.929	0.5849	0.9926
	100	−28.140	52.491	−36.457	9.673	0.5798	0.9828
EDC	1	−1.186	6.0073	−6.235	2.381	0.4985	0.9882
	10	14.148	−14.693	1.735	1.595	0.8423	0.9980
	100	−57.320	80.315	−43.227	9.328	0.7785	0.9863
DC	1	12.390	−8.514	−4.036	3.486	0.5894	0.9963
	10	459.028	−455.404	138.156	−13.985	2.3437	0.9815
	100	289.523	−276.882	72.784	−4.834	1.9677	0.9907

Table 6. Best fittings parameters of Equation (11)(11) $\sigma =a+b\epsilon .$(11) in the region $\epsilon >{\epsilon }_1$and determination coefficients (R²). Engineering stress-Engineering strain.

Chocolate	v (mm/min)	a (MPa)	b (MPa)	R^2
WC	1	0.32 ± 0.03	1.08 ± 0.07	0.9922
	10	0.59 ± 0.01	1.15 ± 0.01	0.9326
	100	1.15 ± 0.02	0.92 ± 0.01	0.9915
RC	1	0.37 ± 0.02	0.95 ± 0.02	0.9819
	10	1.15 ± 0.01	1.36 ± 0.01	0.9870
	100	0.55 ± 0.01	1.58 ± 0.01	0.9750
MC	1	0.95 ± 0.03	1.61 ± 0.08	0.9809
	10	0.57 ± 0.01	2.01 ± 0.03	0.9930
	100	0.83 ± 0.01	3.02 ± 0.02	0.9968
EDC	1	0.80 ± 0.01	2.62 ± 0.10	0.9498
	10	0.320 ± 0.002	1.08 ± 0.05	0.9922
	100	0.587 ± 0.005	1.1 ± 0.4	0.9326
DC	1	1.147 ± 0.006	0.92 ± 0.07	0.9915
	10	0.370 ± 0.002	0.95 ± 0.01	0.9819
	100	1.151 ± 0.008	1.36 ± 0.05	0.9870

Figure 9. Stress vs strain rate for different values of strain.