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Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding

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Abstract

In laser cladding thermal contraction of the initially liquid coating during cooling causes residual stresses and possibly cracks. Preweld or postweld heating using inductors can reduce the thermal strain difference between coating and substrate and thus reduce the resulting stress. The aim of this work is to better understand the influence of various thermometallurgical and mechanical phenomena on stress evolution and to optimize the induction-assisted laser cladding process to get crack-free coatings of hard materials at high feed rates. First, an analytical one-dimensional model is used to visualize the most important features of stress evolution for a Stellite coating on a steel substrate. For more accurate studies, laser cladding is simulated including the powder-beam interaction, the powder catchment by the melt pool, and the self-consistent calculation of temperature field and bead shape. A three-dimensional finite element model and the required equivalent heat sources are derived from the results and used for the transient thermomechanical analysis, taking into account phase transformations and the elastic-plastic material behavior with strain hardening. Results are presented for the influence of process parameters such as feed rate, heat input, and inductor size on the residual stresses at a single bead of Stellite coatings on steel.

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Abbreviations

Symbol:

Quantity

a, mm2/s:

Heat diffusivity

aa, mm:

Ellipsoid half axis (x-direction)

bb, mm:

Ellipsoid half axis (y-direction)

cc, mm:

Ellipsoid half axis (z-direction)

b, mm:

Characteristic length of the bead cross section

c, cP, Ws/(g K):

Specific heat, specific heat of powder particles

d, mm:

Thickness of coating

\({\vec {e}_x,\;\vec {e}_z}\) :

Unit vector along the x-direction, z-direction

erfc(x):

Complementary error function \({\frac{2}{\sqrt{\pi}}\;\int_x^\infty {dx{^\prime}\;\exp \left({-x{^\prime}^{2}} \right)}}\)

f(x, y, z;\({\upalpha_{C}}\)):

Bead shape function

\({\vec {g}\;=\;-g \vec{e}_z}\), mm/s2:

Gravity acceleration vector

h, mm:

Substrate plate thickness

hP(x, y, z; R), Ws/g:

Specific enthalpy of particle with radius R at position (x, y, z)

ierfc(x):

Integrated complementary error function \({\int_x^\infty{dx{^\prime}\;erfc\left({x{^\prime}} \right)}}\)

j(x, y, z), g/(s mm2),:

Mass flow density

j*(x, y, z), g/(s mm2):

Mass flow density with gravity correction

jE(x, y), W/mm2:

Energy flux density of a laser beam incident along the z-direction

j *E (x, y, z), W/mm2:

Local energy flux density of the attenuated laser beam

n :

Scattering exponent

\({\vec {n}}\) :

Outward normal of the workpiece surface

p(x, y, z):

Volume fraction of powder particles in the interaction zone

p*(x, y, z):

Volume fraction of powder particles with gravity correction

q, qL, qI(x, y, z), W/mm3:

Absorbed spatial power density distribution (L: laser, I: induction)

q2,q2L, q2I(x, y), W/mm2:

Absorbed power per unit area (L: laser, I: induction)

\({\vec {q}_{\rm ext}},\) W/mm2:

Total external heat flux density through the workpiece surface

rL, mm:

Laser beam radius

\({\vec {r}},\) mm:

Space vector

\({\vec {r}_{\rm O}},\) mm:

Position of the nozzle outlet

t, s:

Time

\({\vec{v}\;=\;v\;\vec{e}_x},\) mm/s:

Feed rate

v0, m/s:

Initial powder particle speed

w, mm:

Bead width

w(R), mm−1:

Distribution function of powder particle radii

x, mm:

Coordinate along the feed direction

xmax, mm:

Foremost position of the bead/melting zone

xsol(z), mm:

Course of solidification front on the bead surface

y, mm:

Coordinate on the substrate surface across the feed direction

z, mm:

Coordinate perpendicular to the substrate surface

A, mm:

Absorbed fraction of irradiated energy

E, N/mm2:

Young’s modulus

F, mm2:

Area on the substrate surface covered by the powder stream

G(z, t), mm−1:

Green’s function of the one-dimensional heat flow equation

GR(x, y, z), mm−1:

Rosenthal’s solution of heat flow equation for a moving point source

\({\underline{\underline I}}\) :

Second rank unit tensor

J, g/s:

Mass flow (powder rate)

L, mm:

Distance between nozzle outlet and target point

QP, Ws/g:

Latent heat of melting (powder material)

R, m, mm:

Radius (of plate curvature, of particle)

T, °C:

Temperature

T0, °C:

Initial temperature

Tlim, °C:

Minimum required temperature for particles to be resorbed

TM,TMS,TMP, °C:

Melting point temperature, of substrate, of the powder

TP(x, y, z; R), °C:

Temperature of a particle with radius R at position (x, y, z)

TU, °C:

Ambient temperature

Z(x, y ;\({\upalpha_{\rm C}}\)), mm:

Local bead height

\({\upalpha},\) W/(mm2 K):

Heat transfer coefficient

\({\upalpha_{\rm C} \;\equiv \;\upgamma/{\left({g \uprho _0 b^{2}}\right)}}\) :

Capillarity parameter

\({\upbeta}\) :

Geometry parameter

χ:

Shear factor in the ellipsoid model

\({\updelta}\), mm:

Penetration depth of the electromagnetic field

\({-\updelta z\left({\vec {r}}\right),}\) mm:

Local gravity induced correction of particle trajectory

\({\underline{\underline \upvarepsilon},\;\upvarepsilon_{xx}}\) :

Total strain tensor, strain component

\({\underline{\underline \upvarepsilon}^{\rm E},\;\upvarepsilon_{xx}^{\rm E}}\) :

Elastic strain tensor, component

\({\underline{\underline \upvarepsilon}^{\rm I},\;\upvarepsilon_{xx}^{\rm I}}\) :

Inelastic strain tensor, component

\({\underline{\underline \upvarepsilon}^{\rm P},\;\upvarepsilon_{xx}^{\rm P}}\) :

Plastic strain tensor, component

\({\underline{\underline \upvarepsilon}^{\rm T}=\;\upvarepsilon _{\rm T}\;\underline{\underline I}}\) :

Thermometallurgical strain tensor

\({\upgamma}\), N/mm:

Surface tension

\({\upeta}\), mm:

Thickness of the equivalent heat source layer

\({\upkappa}\), mm−1:

Plate curvature

\({\uplambda}\), W/(mm K):

Heat conductivity

\({\upmu}\) :

Relative magnetic permeability

\({\upmu_{0}}\), Vs/(A mm):

Permeability constant

ν:

Poisson’s ratio

\({\uprho}\), V mm/A:

Electrical resistivity

\({\uprho_{0}}\), g/mm3:

Mass density

\({\underline{\underline \upsigma},}\) N/mm2:

Stress tensor

\({\upsigma_{xx},}\) N/mm2:

Longitudinal stress component

\({\upsigma_{yy},}\) N/mm2:

Transverse stress component

\({\upsigma_{\rm Y},}\) N/mm2:

Yield stress

\({\uptau}\), s:

Characteristic time of cooling by convection

\({\upomega}\), s−1:

Circular frequency

\({\Updelta}\)t, \({\Updelta t_{\rm i}}\), s:

Time interval of heating, inductive heating

\({\Updelta}\)m/\({\Updelta x}\), g/mm:

Powder mass deposited per unit length

\({\Updelta z}\), mm:

Interval on the z -axis

\({\Updelta Q_{\rm R}}\), W:

Amount of reflected power

\({\upvartheta}\) :

Scattering angle

\({\Uptheta (x)}\) :

Heaviside’s step function

\({\Upomega}\) :

Space angle

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Acknowledgment

Financial support of this work by the Deutsche Forschungsgemeinschaft within the priority program 1139 is gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Surface Technology and Production Metrology IOF, Dresden University of Technology, Dresden, Germany

    F. Brückner & E. Beyer

  2. Fraunhofer Institute for Material and Beam Technology IWS, Dresden, Germany

    D. Lepski & E. Beyer

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  1. F. Brückner
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Brückner, F., Lepski, D. & Beyer, E. Modeling the Influence of Process Parameters and Additional Heat Sources on Residual Stresses in Laser Cladding. J Therm Spray Tech 16, 355–373 (2007). https://doi.org/10.1007/s11666-007-9026-7

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