A hierarchical asymptotic homogenization approach for viscoelastic composites

Figures & data

Figure 1. (a) Macroscale: viscoelastic heterogeneous material. (b) ${\epsilon }_1$-structural level. Mesoscale: quasi-periodic cell. (c) ${\epsilon }_2$-structural level. Microscale: quasi-periodic cell. The inclusions do not intersect the boundaries.

Figure 2. The hierarchical structure of the dermis similar to Figure 3(v–vi) of Sherman et al. [39]. The figure presents a relation with the Figure 1, where Figure 1 (b) and (c) correspond to Figure 2 (i) and (ii), respectively.

Table 1. Mechanical properties for the constituents of the dermis (see Ref. [27, 37]).

	Young modulus (MPa)	Poisson ratio
Collagen	100	0.48
Ground of substance	10.2	0.48

Table 2. Coefficients of the Prony series for a mice dermis (see Ref. [36]).

	G1	G2	τ1	τ1
Ground of substance	0.3	0.35	0.33	4.59

Figure 3. Computation of the effective relaxation modulus for the dermis. $H/L=0.25$ and $x_2=0.5.$

Table 3. Computation of the effective relaxation modulus and the effective creep compliance for the dermis. The input parameters are given in Tables 1, 2 and Figure 2.

Time (days)	${\widehat{\mathcal{R}}}_{1111}$ (Gpa)	${\widehat{\mathcal{R}}}_{1122}$ (Gpa)	${\widehat{\mathcal{R}}}_{2222}$ (Gpa)	${\widehat{\mathcal{R}}}_{3333}$ (Gpa)	${\widehat{\mathcal{R}}}_{1313}$ (Gpa)	${\widehat{\mathcal{R}}}_{1212}$ (Gpa)
0.8	135.0019	106.9072	112.8124	123.5786	8.3665	4.8150
2.9	134.9209	106.9500	112.6385	123.4420	8.2800	4.6759
5.1	134.9042	106.9588	112.6021	123.4137	8.2621	4.6441
7.2	134.9008	106.9606	112.5948	123.4080	8.2585	4.6377
9.3	134.8996	106.9612	112.5923	123.4060	8.2573	4.6357
11.5	134.8991	106.9615	112.5912	123.4052	8.2568	4.6348
13.6	134.8989	106.9616	112.5907	123.4048	8.2565	4.6343
15.7	134.8987	106.9617	112.5903	123.4045	8.2563	4.6340
17.9	134.8986	106.9617	112.5901	123.4043	8.2562	4.6338
20	134.8986	106.9618	112.5900	123.4042	8.2561	4.6337
time (days)	${\widehat{\mathcal{J}}}_{1111}$ (Gpa)	${\widehat{\mathcal{J}}}_{1122}$ (Gpa)	${\widehat{\mathcal{J}}}_{2222}$ (Gpa)	${\widehat{\mathcal{J}}}_{3333}$ (Gpa)	${\widehat{\mathcal{J}}}_{1313}$ (Gpa)	${\widehat{\mathcal{J}}}_{1212}$ (Gpa)
0.8	0.031562	−0.02154	0.06103	0.04534	0.031020	0.053881
2.9	0.031766	−0.02192	0.06215	0.04585	0.031324	0.055452
5.1	0.031806	−0.02201	0.06239	0.04594	0.031379	0.055813
7.2	0.031815	−0.02202	0.06243	0.04596	0.031391	0.055885
9.3	0.031817	−0.02203	0.06245	0.04597	0.031395	0.055908
11.5	0.031819	−0.02203	0.06246	0.04597	0.031396	0.055918
13.6	0.031819	−0.02203	0.06246	0.04597	0.031397	0.055923
15.7	0.03182	−0.02203	0.06246	0.04597	0.031398	0.055926
17.9	0.03182	−0.02203	0.06246	0.04597	0.031398	0.055928
20	0.03182	−0.02203	0.06246	0.04597	0.031398	0.05593

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