A modification of the semi-analytic inversion method: determination of the yield stress and a comparison with the parametrization algorithm

Figures & data

Fig. 1 The schematical illustration of the Algorithms 1 and 2.

Fig. 2 The function $g({\mathit{\xi }}^2)$ defined by (6).

Table 1 Synthetic noise free data for the stiff and the soft materials .

$E=210$ Gpa	$m=1$	$m=2$	$m=3$	$m=4$	$m=5$
${\mathit{\phi }}_m\times {10}^3$	2.2	3.3	3.6	3.7	3.85
${\mathit{\xi }}_m^2\times {10}^2$	1.3	4.0	6.6	7.9	10.0
${\mathcal{T}}_m\times {10}^2$	2.48	3.75	4.21	4.41	4.7
$E=110$ Gpa	$m=1$	$m=2$	$m=3$	$m=4$	$m=5$
${\mathit{\phi }}_m\times {10}^3$	4.4	5.58	5.97	6.28	6.6
${\mathit{\xi }}_m^2\times {10}^2$	1.4	3.6	5.2	7.1	9.8
${\mathcal{T}}_m\times {10}^2$	2.59	3.35	3.68	4.0	4.39

Fig. 3 The reconstructed curve $g_h({\mathit{\xi }}^2)$ for different values of angle steps for the stiff (left figure) and the soft (right figure) materials, (noise free data).

Table 2 The approximate values of $\mathit{\kappa }$ corresponding to each plastic case for the stiff material.

$(\mathit{\phi }\times {10}^3,\mathcal{T}\times {10}^2)$	$\mathrm{\Delta }\mathit{\phi }\times {10}^3$	$g_h({\mathit{\xi }}^2)$	${\mathit{\xi }}_{1h}^2$	$g_h({\mathit{\xi }}_{1h}^2)$	${\mathit{\kappa }}_h$
$(3.30,3.75)$	$1.1$	$0.0124-0.1385({\mathit{\xi }}^2-0.0252)$	0.030	0.0117	0.35
$(3.60,4.21)$	$1.4$	$0.0124-0.0900({\mathit{\xi }}^2-0.0252)$	0.058	0.0094	0.34
$(3.70,4.41)$	$1.5$	$0.0124-0.0800({\mathit{\xi }}^2-0.0252)$	0.070	0.0088	0.33
$(3.85,4.71)$	$1.65$	$0.0124-0.0670({\mathit{\xi }}^2-0.0252)$	0.092	0.0073	0.30

Table 3 The approximate values of $\mathit{\kappa }$ corresponding to each plastic case for the soft material.

$(\mathit{\phi }\times {10}^3,\mathcal{T}\times {10}^2)$	$\mathrm{\Delta }\mathit{\phi }\times {10}^3$	$g_h({\mathit{\xi }}^2)$	${\mathit{\xi }}_{1h}^2$	$g_h({\mathit{\xi }}_{1h}^2)$	${\mathit{\kappa }}_h$
$(5.58,3.35)$	$1.18$	$0.0238-0.2900({\mathit{\xi }}^2-0.0185)$	0.033	0.0196	0.33
$(5.97,3.68)$	$1.57$	$0.0238-0.2230({\mathit{\xi }}^2-0.0185)$	0.048	0.0172	0.31
$(6.28,4.00)$	$1.88$	$0.0238-0.1800({\mathit{\xi }}^2-0.0185)$	0.066	0.0153	0.30
$(6.60,4.39)$	$2.20$	$0.0238-0.1390({\mathit{\xi }}^2-0.0185)$	0.095	0.0130	0.27

Table 4 The absolute and relative errors for the noise free inverse problem.

$E=210$ Gpa	Paramet.	Fast alg.
${\mathit{ϵ}}_g$	$4.4\times {10}^{-4}$	$1.7\times {10}^{-4}$
${\mathit{\delta }}_g$	$6.0\times {10}^{-2}$	$1.5\times {10}^{-2}$
$E=110$ Gpa	Paramet.	Fast alg.
${\mathit{ϵ}}_g$	$4.6\times {10}^{-4}$	$4.3\times {10}^{-4}$
${\mathit{\delta }}_g$	$4.2\times {10}^{-2}$	$1.9\times {10}^{-2}$

Fig. 4 The reconstructed curve $g_h({\mathit{\xi }}^2)$ for $\mathrm{\Delta }\mathit{\phi }=1.65\times {10}^{-3}$ for the stiff material (left figure ) and the reconstructed curve $g_h({\mathit{\xi }}^2)$ for $\mathrm{\Delta }\mathit{\phi }=2.20\times {10}^{-3}$ for the soft material (right figure), (noise data).

Table 5 The absolute and relative errors for the noise inverse problem.

$\mathit{\gamma }$ (For stiff)	${\mathit{ϵ}}_g$ (Paramet.)	${\mathit{ϵ}}_g$ (Fast alg.)	${\mathit{\delta }}_g$ (Paramet.)	${\mathit{\delta }}_g$ (Fast alg.)
$+0.03$	$9.3\times {10}^{-4}$	$8.5\times {10}^{-4}$	$1.3\times {10}^{-1}$	$1.0\times {10}^{-1}$
$-0.05$	$2.0\times {10}^{-3}$	$1.0\times {10}^{-3}$	$2.7\times {10}^{-1}$	$1.9\times {10}^{-1}$
$\pm 0.07$	$2.2\times {10}^{-3}$	$1.3\times {10}^{-3}$	$2.8\times {10}^{-1}$	$2.1\times {10}^{-1}$
$\mathit{\gamma }$ (For soft)	${\mathit{ϵ}}_g$ (Paramet.)	${\mathit{ϵ}}_g$ (Fast alg.)	${\mathit{\delta }}_g$ (Paramet.)	${\mathit{\delta }}_g$ (Fast alg.)
$+0.03$	$1.2\times {10}^{-3}$	$1.0\times {10}^{-3}$	$9.3\times {10}^{-2}$	$8.6\times {10}^{-2}$
$-0.05$	$2.5\times {10}^{-3}$	$2.3\times {10}^{-3}$	$2.0\times {10}^{-1}$	$1.7\times {10}^{-1}$
$\pm 0.07$	$2.6\times {10}^{-3}$	$2.4\times {10}^{-3}$	$2.2\times {10}^{-1}$	$1.9\times {10}^{-1}$