Mechanics of Composite Materials

, Volume 49, Issue 2, pp 221–230 | Cite as

Springback Angle of a C/PPS Laminate with a Textile Reinforcement


The residual stresses arising in fiber-reinforced laminates during their curing in closed molds lead to changes in the composites after their removal from the molds and cooling. One of these dimensional changes of angle sections is called springback. The article compares the springback angles computed by a model representating the weave geometry (for plain and satin weaves) and by a model with straight fibers with values measured after the manufacturing process. A comparison between the thermoelastic characteristics of composites computed by both the models also presented.


springback thermoplastic matrix carbon fiber reinforcement woven composites 


The use of hi-tech thermoplastic matrices (e.g., PEKK, PEEK, or PPS) in carbon-fiber-reinforced composites constantly grows, especially in the aircraft industry. In addition, thermoplastic materials show significant advantages during their fabrication and allow the application of an optimized metal processing technology (stamping). However, the high temperature at which a thermoplastic composite must be processed suggests an increased significance of thermally induced stresses and distortions in a finished product. This is why the prediction of its dimensional changes (e.g., the springback) is necessary to make the final part more precise.

The change in composite dimensions is related to many parameters: part angles, thicknesses, lay-ups, flange length, and also tool materials, tool surface, and cure cycles [1]. When a composite part in the shape of L (or U, Fig. 1) section is extracted from a mold after cooling to room temperature, a change in the angle of the part can be observed.
Fig. 1

Distortion of a molded U section.

Tool angles have to be modified to eliminate this problem. The tool design is based on either the rules of thumb, on the past experience, or on the trial-and-error approach. For angular parts, the compensation is normally between 1 and 3°. The most common problem encountered is the fact that the springback may vary with lay-up, material, cure temperature, etc. Therefore, what worked once will not necessarily work next time.

1. Representation of Weave Geometry

Because of the regular structure of a woven composite, the thermoelastic properties of the entire composite are similar to the properties of its typical element. For calculating these properties, we have to know
  • the mass of fabric M, g/m2,

  • the number of threads n, 1/cm,

  • fabric thickness h, mm,

  • the warp and weft materials and their geometry, and

  • weave type.

The most common types of weaves are plain, twill, and satin ones (see Fig. 2.). In each weave type, a regularly repeating area can be found. Three types repeating elements can be distinguished in the weaves (see Fig. 3).
Fig. 2

Weave types: plain (a), twill (b), and satin (c).


Dimensions and types of elements [2].

In element I, both fibers are undulated (typical of the plain weave); in element II, both fibers are straight; in element III, at least one fiber is curved. It can be shown that the twill and satin weaves can be composed of elements I and II [2]. Element I is the basic one, because the other ones are its special cases. To calculate the properties of element I, we introduce the following presumptions:
  • the total thickness of the element is the sum of fabric and matrix thicknesses;

  • the weave of fabric is tight;

  • the fibers are prismatic, and their curvature is regular (sinusoidal);

  • fibers in the cross section of the element are distributed equally;

  • the matrix and fibers are linearly elastic, the matrix is isotropic, and the fibers are transversely isotropic;

  • the temperature is the same throughout the volume of the element, and residual stresses are absent;

  • no other constituents or defects exist in the composite along with the fibers and matrix.

The effective deformation characteristics of the element can be expressed as
$$ {S_{ij }}=\frac{1}{{2\varOmega }}\int\nolimits_{{-\varOmega}}^{\varOmega } {{S_{ij }}(\omega )d\omega, } $$
where S ij are elements of the compliance matrix, ω is the undulation angle of warp (or weft) threads, and and Ω is the maximum value of ω, which depends on the thickness and width of threads [2],
$$ \begin{array}{*{20}{c}} {{\varOmega_i}=\arctan \frac{{\pi {h_i}}}{{2{t_i}}},} & {i=1,2.} \\ \end{array} $$
Here, t i and h i are the width and thickness of weft threads (i = 1, 2 corresponds to the warp and weft directions, respectively);
$$ \begin{array}{*{20}{c}} {{t_2}=\frac{100 }{{{n_y}}},} & {{h_2}=h-{h_1},} \\ \end{array} $$
where n y is the number of weft threads (the number 100 in the numerator stands for 100 mm, because the number of threads in the material sheet is given on 10 cm, but our model calculates with the number of threads on 1 cm), h is the total thickness of the fabric, and h 1 is the thickness of warp threads,
$$ {h_1}=\frac{{\xi {t_2}h}}{{{t_1}+\xi {t_2}.}} $$
Here ξ is the ratio between the total cross-sectional areas of warp and weft threads (ξ = 1 when the number of warp and weft threads is the same), and t 1 is the width of the warp threads.
$$ {t_1}=\frac{100 }{{{n_x}}}, $$
where n x is the number of warp threads. Since we focus exactly on the influence of the angle, it can be assumed that the matrix thickness h 3 is constant (h 3 = 0).
It can be shown that the deformation properties of a composite made from a twill or satin lies between those of elements I and II. This means that its compliance matrix \( \bar{S} \) is the weighted average of those of elements I and II (S I and S II):
$$ \bar{S}=\frac{{i{S_{\mathrm{I}}}+j{S_{\mathrm{I}\mathrm{I}}}}}{i+j }, $$
where i and j are the numbers of elements I and II.
The representation of weave geometry [2] is done for the warp thread (superscript 1) with the orientation angle θ = 0°. After integration according to Eq. (1) and replacement of the goniometric functions with the first terms of their Taylor series, we obtain the following effective characteristics for the warp layer:
$$ \begin{array}{*{20}{c}} {^1{E_x}=\frac{{{E_L}}}{{1+\frac{{{\varOmega^2}}}{3}\left[ {\frac{{{E_L}}}{{{G_{LT }}}}-2\left( {1+{\nu_{LT }}} \right)} \right]}},} \\ {^1{E_y}={E_T},} \\ {^1{\nu_{yx }}={\nu_{TL }}+\left( {{\nu_{{T{T}^{\prime}}}}-{\nu_{TL }}} \right)\frac{{{\varOmega^2}}}{3},} \\ {^1{G_{xy }}=\frac{{{G_{LT }}}}{{1+\frac{{{\varOmega^2}}}{3}\left( {\frac{{{G_{LT }}}}{{{G_{{T{T}^{\prime}}}}}}-1} \right)}},} \\ {^1{\alpha_x}={\alpha_L}+\frac{{{\varOmega^2}}}{3}\left( {{\alpha_T}-{\alpha_L}} \right),} \\ {^1{\alpha_y}={\alpha_T},} \\ {^1{E_z}=\frac{{{E_T}}}{{1+\frac{{{\varOmega^2}}}{3}\left[ {\frac{{{E_T}}}{{{G_{LT }}}}-2\left( {1+{\nu_{TL }}} \right)} \right]}},} \\ {^1{G_{xz }}=\frac{{{G_{LT }}}}{{\left[ {\frac{{{G_{LT }}}}{{{E_L}}}\left( {1+2{\nu_{LT }}} \right)+\frac{{{G_{LT }}}}{{{E_T}}}-1} \right]\frac{{4{\varOmega^2}}}{3}+1}},} \\ {^1{G_{yz }}=\frac{{{G_{{T{T}^{\prime}}}}}}{{1+\frac{{{\varOmega^2}}}{3}\left[ {\frac{{{G_{{T{T}^{\prime}}}}}}{{{G_{LT }}}}-1} \right]}},} \\ {^1{\nu_{yz }}={\nu_{{T{T}^{\prime}}}}+\left( {{\nu_{TL }}-{\nu_{{T{T}^{\prime}}}}} \right)\frac{{{\varOmega^2}}}{3},} \\ \end{array} $$
$$ \begin{array}{*{20}{c}} {^1{\nu_{xz }}={E_x}\left[ {\frac{{{\upsilon_{LT }}}}{{{E_L}}}-\left( {\frac{1}{{{E_L}}}+\frac{1}{{{E_T}}}+\frac{{2{\upsilon_{LT }}}}{{{E_L}}}-\frac{1}{{{G_{LT }}}}} \right)\frac{{{\varOmega^2}}}{3}} \right],} \\ {^1{\alpha_z}={\alpha_T}+\frac{{{\varOmega^2}}}{3}\left( {{\alpha_L}-{\alpha_T}} \right),} \\ \end{array} $$
where 1 E, 1 G, 1ν, and 1 α are Young’s modulus, the shear modulus, Poisson’s ratio, and the thermal expansion coefficient; the subscripts x, y, and z are related to the coordinate system according to Fig. 1. These effective characteristics are computed from the characteristics for straight fibers (where the subscripts L, T, and T′ stand for the longitudinal, transverse, and throughthickness directions, and Ω is the undulation angle of warp). Similarly, we can express characteristics for the weft threads (superscript 2), i.e., θ = 90°. For orientation angles θ ≠ (0 and 90°), the elastic characteristics are computed from the relation
$$ S=\left[ {\begin{array}{*{20}{c}} {\frac{1}{{{E_x}}}} \hfill & {-\frac{{{\nu_{xy }}}}{{{E_x}}}} \hfill & {-\frac{{{\nu_{zx }}}}{{{E_z}}}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {-\frac{{{\nu_{xy }}}}{{{E_x}}}} \hfill & {\frac{1}{{{E_y}}}} \hfill & {-\frac{{{\nu_{zy }}}}{{{E_z}}}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {-\frac{{{\nu_{zx }}}}{{{E_z}}}} \hfill & {-\frac{{{\nu_{zy }}}}{{{E_z}}}} \hfill & {\frac{1}{{{E_z}}}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {\frac{1}{{{G_{{T{T}^{\prime}}}}}}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{1}{{{G_{{L{T}^{\prime}}}}}}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {\frac{1}{{{G_{LT }}}}} \hfill \\ \end{array}} \right], $$

But the compliances are found from the well-known transformation relations [4, 5]. These properties are used to calculate the effective thermoelastic characteristic by using the classical lamination theory (CLT) and equations for the throughthickness characteristics of composite plates of the given lay-up (see [4] or [5] for details).

2. Micromechanics of Straight Fibers

The difference between straight and undulated fibers is in the value of the angle Ω (which is equal to 0° for straight fibers). Thus, at Ω = 0°, Eqs. (2) give the values of thermoelastic characteristics of a unidirectional composite plate with straight fibers. Knowing the fiber volume fraction V f , we can calculate the basic thermoelastic constants according to [3]:
$$ \begin{array}{*{20}{c}} {{E_L}={V_f}{E_{fL }}+\left( {1-{V_f}} \right){E_m},} \\ {{E_T}=\frac{{{E_m}}}{{1-\sqrt{{{V_f}}}\left( {1-\frac{{{E_m}}}{{{E_{fT }}}}} \right)}},} \\ {{E_{{T^{\prime}}}}={E_T},} \\ {{G_{LT }}=\frac{{{G_m}}}{{1-\sqrt{{{V_f}}}\left( {1-\frac{{{G_m}}}{{{G_{f12 }}}}} \right)}},} \\ {{G_{{T{T}^{\prime}}}}=\frac{{{G_m}}}{{1-\sqrt{{{V_f}}}\left( {1-\frac{{{G_m}}}{{{G_{f23 }}}}} \right)}},} \\ {{G_{{L{T}^{\prime}}}}={G_{LT }},} \\ {{\nu_{LT }}={V_f}{\nu_f}+\left( {1-{V_f}} \right){\nu_m},} \\ {{\nu_{{T{T}^{\prime}}}}=\frac{{{E_T}}}{{{G_{{T{T}^{\prime}}}}}}-1,} \\ {{\nu_{{L{T}^{\prime}}}}={\nu_{LT }},} \\ {{\alpha_L}=\frac{{{V_f}{\alpha_{fL }}{E_{fL }}+\left( {1-{V_f}} \right){\alpha_m}{E_m}}}{{{V_f}{E_{fL }}+\left( {1-{V_f}} \right){E_m}}},} \\ {{\alpha_T}={\alpha_{fT }}\sqrt{{{V_f}}}+\left( {1-\sqrt{{{V_f}}}} \right)\left( {1+\frac{{{V_f}{\nu_m}{E_m}}}{{{E_L}}}} \right){\alpha_m},} \\ {{\alpha_{{T^{\prime}}}}={\alpha_T},} \\ \end{array} $$
where the subscripts L, T, and T′ designate the longitudinal, transverse, and through-thickness directions, and f and m stand for the fiber and matrix, respectively.

3. Springback Phenomenon

The springback is an angular change in a composite which occurs upon releasing a laminated part from the mold and cooling. This effect depends on many factors, which are mentioned at the beginning of the article. The main factors considered in analytical computations are the thermal expansion in laminates, the shrinkage of resin during the curing process (this effect is relevant only to semicrystalline matrices, which transform from amorphous to crystalline phases during the cure cycle), and moisture absorption. The total change in the angle can be written as the sum
$$ \varDelta \gamma =\varDelta {\gamma_t}+\varDelta {\gamma_h}+\varDelta {\gamma_c}=\gamma \frac{{\varepsilon_y^t-\varepsilon_z^t}}{{1+\varepsilon_z^t}}+\gamma \frac{{\varepsilon_y^h-\varepsilon_z^h}}{{1+\varepsilon_z^h}}+\gamma \frac{{\varepsilon_y^c-\varepsilon_z^c}}{{1+\varepsilon_z^c}}, $$
where Δγ t , Δγ h , and Δγ c are the changes in the angle caused by temperature, the hygroscopic effect, and shrinkage, respectively; ε y and ε z are strains in the longitudinal and thickness directions [6]. These strains can be computed using equations of the CLT in combination with equations for the through-thickness characteristics [4]. The final equations are
$$ \left\{ {\begin{array}{*{20}{c}} {N_x^{thc }} \\ {N_y^{thc }} \\ {N_{xy}^{thc }} \\ {M_x^{thc }} \\ {M_y^{thc }} \\ {M_{xy}^{thc }} \\ \end{array}} \right\}=\left[ {\begin{array}{*{20}{c}} {{A_{11 }}} & {{A_{12 }}} & {{A_{16 }}} & {{B_{11 }}} & {{B_{12 }}} & {{B_{16 }}} \\ {{A_{21 }}} & {{A_{22 }}} & {{A_{26 }}} & {{B_{21 }}} & {{B_{22 }}} & {{B_{26 }}} \\ {{A_{61 }}} & {{A_{62 }}} & {{A_{66 }}} & {{B_{61 }}} & {{B_{62 }}} & {{A_{66 }}} \\ {{B_{11 }}} & {{B_{12 }}} & {{B_{16 }}} & {{D_{11 }}} & {{D_{12 }}} & {{D_{16 }}} \\ {{B_{12 }}} & {{B_{22 }}} & {{B_{26 }}} & {{D_{21 }}} & {{D_{22 }}} & {{D_{26 }}} \\ {{B_{61 }}} & {{B_{62 }}} & {{B_{66 }}} & {{D_{61 }}} & {{D_{62 }}} & {{D_{66 }}} \\ \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\varepsilon_x^{0,thc }} \\ {\varepsilon_y^{thc }} \\ {\gamma_{xy}^{0,thc }} \\ {\kappa_x^{thc }} \\ {\kappa_y^{thc }} \\ {\kappa_{xy}^{thc }} \\ \end{array}} \right\}, $$
where A ij , B ij , and D ij are the generally known elements of membrane stiffness, bending-extension coupling stiffness, and bending stiffness matrices; the quantities \( N_i^{thc } \) and \( M_i^{thc } \) are defined by the equations
$$ \begin{array}{*{20}{c}} {N_i^{thc }=\int {{Q_{ij }}\varepsilon_j^{thc }dz,} } \\ {M_i^{thc }=\int {{Q_{ij }}\varepsilon_j^{thc }zdz}, } \\ \end{array} $$
where the integration is from –H/2 to H/2. \( N_i^{thc } \) and \( M_i^{thc } \) have the same dimension as N i and M i and are called the resultants of internal thermohygrocrystallic forces and moments; \( \varepsilon_{ij}^{0,thc } \) and \( \kappa_{ij}^{0,thc } \) are the strains and curvatures. Q is the 2D plane-stress matrix in the (L, T) coordinate system (see Fig.1).
The through-thickness strain can be computed as
$$ \varepsilon_z^{thc }=\frac{{\varDelta {H^{thc }}}}{H}, $$
where H is the total thickness of the composite part, and ΔH thc is the change in the thickness due to the thermohygrocrystallic effect,
$$ \varDelta {H^{thc }}=\varDelta {H^t}+\varDelta {H^h}+\varDelta {H^c}. $$
The thermal, hygral, and crystallic changes in thickness can be computed as
$$ \begin{array}{*{20}{c}} {\varDelta {H^t}=\sum\limits_{k=1}^N {\left\{ {\left[ {{S_{13 }}{S_{23 }}{S_{36 }}} \right]{{{\left[ {{T_q}} \right]}}_k}\left[ {\left( {{z_k}-{z_{k-1 }}} \right)\left[ {\overline{{{Q_k}}}} \right]\left( {\left[ {\begin{array}{*{20}{c}} {\varepsilon_x^0} \\ {\varepsilon_y^0} \\ {\gamma_{xy}^0} \\ \end{array}} \right]-\varDelta T{{{\left[ {\begin{array}{*{20}{c}} {{\alpha_x}} \\ {{\alpha_y}} \\ {{\alpha_{xy }}} \\ \end{array}} \right]}}_k}} \right)} \right.} \right.} } \\ {\left. {\left. {+\frac{{z_k^2-z_{k-1}^2}}{2}\left[ {\overline{{{Q_k}}}} \right]\left[ {\begin{array}{*{20}{c}} {{\kappa_x}} \\ {{\kappa_y}} \\ {{\kappa_{xy }}} \\ \end{array}} \right]} \right]+\left( {{z_k}-{z_{k-1 }}} \right)\left[ {\varDelta T{{{\left( {{\alpha_3}} \right)}}_k}} \right]} \right\},} \\ {\varDelta {H^h}=\sum\limits_{k=1}^N {\left\{ {\left[ {{S_{13 }}{S_{23 }}{S_{36 }}} \right]{{{\left[ {{T_q}} \right]}}_k}\left[ {\left( {{z_k}-{z_{k-1 }}} \right)\left[ {\overline{{{Q_k}}}} \right]\left( {\left[ {\begin{array}{*{20}{c}} {\varepsilon_x^0} \\ {\varepsilon_y^0} \\ {\gamma_{xy}^0} \\ \end{array}} \right]-\varDelta c{{{\left[ {\begin{array}{*{20}{c}} {{\beta_x}} \\ {{\beta_y}} \\ {{\beta_{xy }}} \\ \end{array}} \right]}}_k}} \right)} \right.} \right.} } \\ {\left. {\left. {+\frac{{z_k^2-z_{k-1}^2}}{2}\left[ {\overline{{{Q_k}}}} \right]\left[ {\begin{array}{*{20}{c}} {{\kappa_x}} \\ {{\kappa_y}} \\ {{\kappa_{xy }}} \\ \end{array}} \right]} \right]+\left( {{z_k}-{z_{k-1 }}} \right)\left[ {\varDelta c{{{\left( {{\beta_3}} \right)}}_k}} \right]} \right\},} \\ {\varDelta {H^c}=\sum\limits_{k=1}^N {\left\{ {\left[ {{S_{13 }}{S_{23 }}{S_{36 }}} \right]{{{\left[ {{T_q}} \right]}}_k}\left[ {\left( {{z_k}-{z_{k-1 }}} \right)\left[ {\overline{{{Q_k}}}} \right]\left( {\left[ {\begin{array}{*{20}{c}} {\varepsilon_x^0} \\ {\varepsilon_y^0} \\ {\gamma_{xy}^0} \\ \end{array}} \right]-{{{\left[ {\begin{array}{*{20}{c}} {{\varphi_x}} \\ {{\varphi_y}} \\ {{\varphi_{xy }}} \\ \end{array}} \right]}}_k}} \right)+\frac{{z_k^2-z_{k-1}^2}}{2}\left[ {\overline{{{Q_k}}}} \right]\left[ {\begin{array}{*{20}{c}} {{\kappa_x}} \\ {{\kappa_y}} \\ {{\kappa_{xy }}} \\ \end{array}} \right]} \right]+\left( {{z_k}-{z_{k-1 }}} \right){{{\left( {{\varphi_3}} \right)}}_k}} \right\},} } \\ \end{array} $$
where N is the number of layers, S ij are elements of the compliance matrix, T q is the transformation matrix, z is the coordinate of a layer, \( \bar{Q} \) is the 2D stress matrix in the (x, y) coordinate system (see Fig. 1), ΔT is the change in temperature, Δc is the change in moisture, α ij is the thermal expansion coefficient, β ij is the coefficient of moisture absorption, and φ ij is the coefficient of chemical shrinkage during recrystallization.
For parts with a single curvature (which are analyzed in our case), Eq. (3) is transformed:
$$ \kappa_y^{thc}\Rightarrow \kappa_y^{thc }+\frac{1}{{{R_y}}}\left( {\varepsilon_y^{thc }-\varepsilon_z^{thc }} \right), $$
where R y is the radius of the analyzed part in the y direction.

4. Comparison of Models

To compare the springback angles computed by both models, we used a C/PPS composite material (manufactured by Letov Letecká Výroba, s.r.o.) reinforced with a 5H satin weave, with a fiber volume content V f = 49%, mass M = 285 g/m2, fabric, Toray T300J 3K fibers, and thread number n x = n y = 70 bundles per 10 cm for the warp and weft threads. The thickness of the fabric was 0.3 mm. The thermoelastic characteristics of the fibers and matrix were as follows: E fL = 230 GPa, E fT = 15 GPa, ν f = 0.3, G f12, = 50 GPa, G f23 = 27 GPa*, α fL = −3.8·10–7 °C−1, α fT = 12.5·10−6 °C–1*, Φ f = 0, E m = 3800 MPa, ν m = 0.36, α m = 5.2·10−5 °C–1, and Φ m = 2.015%; the data with asterisks are estimated according to [3]; ΔT = 160°C and Δc = 0.

The results obtained are compared with those for a plain weave in Fig. 4 (the coefficients of chemical shrinkage are not plotted, because their behavior is similar to that of thermal expansion coefficients). Since the lay-ups are symmetric, their thermoelastic properties in the x and y directions are equal. On the horizontal axes of the diagrams, the undulation angle of fibers Ω is plotted. On the vertical axes, ratios between the characteristics of composites with undulated and straight fibers are laid off. The black squares denote the values computed for plates with a single curvature.
Fig. 4

The quantities α z /α z0 (a), α x /α x0 (b), E x /E x0 (c) G xy /G xy0 (d), ν xy xy0 (e), and SB/SB 0 (f) as functions of the angle Ω for 16-layer [[(0,90)/(±45)]4]s (1), 18-layer [[(0,90)/(±45)]4/(0,90)]s (2), and 20-layer [(0,90)/[(0,90)/(±45)]4/(0,90)]s (3) laminates with plane (––––) and satin-weave (– – –) fabrics.

From Fig. 4, it is seen that a z , E x , and G xy and the springback angle SB decrease with increasing fiber undulation angle for all the lay-ups and both the weaves considered. However, α x , and n xy increase with the angle in all these cases lay-ups and for both the weaves analyzed. A detailed comparison of springback angles SB for the given composite plates is presented in Fig. 5. The average values and standard deviation for the 16-, 18-, and 20-layer laminates were computed from 25, 10, and 10 measurements, respectively.
Fig. 5

Comparison of springback angles computed for the plates investigated.

5. Conclusions

A springback analysis of curved C/PPS composite parts with textile reinforcement has been performed. An analytical model with straight fibers was used and compared with a model with undulated fibers, which can be found in woven textile composites. The thermoelastic characteristics of various layered composites in relation to the fiber undulation angle were constructed.

It the case of a satin-weave fabric 0.3 mm thick, the maximum difference between the thermoelastic characteristics computed by the models with straight and undulated fibers was 4% (for the moduli E x and G xy ). With a plain-weave fabric, the maximum difference would be greater — 12% (again for E x and G xy ). For the given composite parts, for all the lay-ups considered (for straight fibers and plain and satin weaves), the computed values fit into the range limited by the standard deviation.

The analytical model presented can also be used for hybrid fabrics.


Acknowledgements. This work was supported by the Ministry of Industry and Trade of the Czech Republic number FR-TI1/463 and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS12/176/OHK2/3T/12.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department. of Mechanics, Biomechanics and MechatronicsFaculty of Mechanical Engineering, CTUPragueCzech Republic

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