Latest revision as of 20:18, 17 November 2022

1 Introducción

Los componentes de materiales compuestos sufren, en general, distorsiones significativas durante el proceso de fabricación. Estas distorsiones tienen diversas causas [1], siendo las más destacadas, la anisotropía del comportamiento termoelástico, la contracción que sufre la resina durante el curado y la interacción entre la pieza y el molde.

En los componentes típicos de la industria aeronáutica, formados mediante apilados de láminas de fibra de carbono y resina epoxi, la deformación térmica del componente está muy influenciada por la deformación sufrida en los codos que se forman al doblar algunos laminados en las zonas de unión [2].

El objetivo del presente trabajo es caracterizar el comportamiento termoelástico de un angular fabricado con un laminado de material compuesto, con objeto de determinar sus coeficientes de expansión térmica CET, tanto dentro del plano como en la dirección del espesor.

Radford y Diefendorf [3] emplearon una probeta angular en forma de L para determinar el CET en la dirección del espesor, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{z}} , a partir de la distorsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}

 sufrida por la misma al ser sometida a un incremento térmico, ver Figura 1. Para determinar la distorsión midieron la desviación que sufría un rayo láser reflejado sobre un espejo adherido a una de las caras de la probeta.

La relación entre el incremento térmico Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta T}

sufrido por la probeta y la distorsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}
 viene dada por [3]:

siendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta}

 el ángulo interior del codo y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}}
el CET en la dirección afectada por la curvatura (dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
en la Figura 1).

Recientemente, otros autores han empleado técnicas de interferometría [4] o videocorrelación [5] para determinar los CET dentro del plano en probetas planas.

En el presente trabajo se combinan ambas técnicas de forma que, mediante videocorrelación, se determinan los desplazamientos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{x}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{y}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}}
de la cara frontal de una probeta en L (como la mostrada en la Figura 1, donde se indica la referencia empleada para las coordenadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

) sometida a un incremento de temperatura. A partir de estas medidas, se pueden determinar simultáneamente los CET Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{x}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}}
en el plano de dicha cara y la distorsión Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}
 que sufre la probeta. A partir de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}
 se determina Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{z}}
empleando (1).

2 Ensayos

Se han fabricado tres probetas en forma de L, con laminados de 8 láminas de preimpregnados de fibra carbono y resina epoxi (Hexply AS4 / 8552 [6]). Las secuencias de apilado empleadas han sido:

• Probeta 0: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{0\mbox{°}}_{8}]}

• Probeta 90: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{90\mbox{°}}_{8}]}

• Probeta QI: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {[90\mbox{°},45\mbox{°},-45\mbox{°},0\mbox{°}]}_{S}}

En las láminas orientadas a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\mbox{°}}

las fibras siguen la curvatura de la probeta, es decir, son paralelas a la dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
mostrada en la Figura 1.

Las caras planas de las probetas en L tienen dimensiones de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \, 60\, mm\, \times \, 30\, mm} , el espesor es de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1.6\, mm}

y el radio interior de la zona curva de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 5\, mm}

.

Para determinar los desplazamientos sufridos por la probeta durante el calentamiento de las mismas, se han introducido en un horno, con puerta trasparente, de forma que una de sus caras quede paralela a la puerta del horno (ver Figura 2). Los desplazamientos se han medido empleando un sistema de videocorrelación en tres dimensiones con dos cámaras Tele-dyne Imaging Falcon 2 12M (4096 × 3072 @ 8 fps). Para capturar el movimiento de las caras de las probetas enfrentadas al equipo de videocorrelación, éstas se han marcado con un punteado en blanco y negro.

Para comprobar que la temperatura del horno está estabilizada antes de realizar cada medida, se ha fijado un termopar en la base de una de las probetas.

3 Medidas

Las medidas se han llevado a cabo sobre una zona rectangular de aproximadamente Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 20\, mm}

de ancho por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 40\, mm}
de alto. El software del sistema de videocorrelación VIC-3D permite determinar los desplazamientos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{x}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{y}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}}
en una rejilla situada en dicha zona. Adicionalmente, ofrece también las deformaciones en el plano de la probeta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xy}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}}

.

Para determinar los CET Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{x}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}}
dentro del plano del laminado se emplean las medidas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}}

, mientras que para determinar el CET Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{z}}

en la dirección del espesor se emplean las medidas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}}

.

3.1 Deformaciones dentro del plano

En las Figuras 3(a,b) se observan (a modo de ejemplo) las deformaciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}}
proporcionadas por el equipo de videocorrelación, para una temperatura T=120ºC, en la zona de medida de la probeta 0. Como puede observarse, dado que la variación térmica que sufre la probeta es uniforme, la deformación en la cara de la probeta es razonablemente uniforme. Los puntos negros que se aprecian en las gráficas corresponden a la retícula generada por el equipo de postproceso, en la que se obtienen valores numéricos de las medidas. Tomando el valor medio de las medidas en dichos puntos se obtienen los resultados mostrados en la Tabla 1. La temperatura ambiente al comenzar el proceso era de Ta=26ºC. Se han medido las deformaciones, a una temperatura ligeramente superior a la temperatura ambiente, y a intervalos aproximadamente regulares de 30ºC en 30ºC.

[ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu m}

[ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu m}

 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  ^oC}

3.2 Desplazamientos fuera del plano

En la Figuras 3(c) se observan (a modo de ejemplo) los desplazamientos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}}

proporcionados por el equipo de videocorrelación, para una temperatura T=120 ºC, en la zona de medida de la probeta 0. Como puede observarse, dado que la cara de la probeta sufre un giro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}
 (como el indicado en la Figura 1), los desplazamientos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}}
aumentan linealmente con la coordenada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

. Para cada temperatura, la relación entre los desplazamientos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}(y)}

y el ángulo girado por la cara viene dada por:

La relación lineal entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {u}_{z}}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
se observa claramente en la Figura 4, en la que se muestran las medidas obtenidas, para cada una de las probetas y temperaturas, en todos los puntos de la retícula.

Haciendo un ajuste lineal de cada una de las gráficas con el método de los mínimos cuadrados, se obtienen las medidas del ángulo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}

 girado por cada una de las probetas en función de la temperatura, mostrados en la Tabla 2.

 [ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mbox{°}}

[ºC]

4 Coeficientes de expansión térmica

Como se ha dicho anteriormente, a partir de las medidas de las deformaciones dentro del plano en la cara de la probeta es posible determinar los CET dentro del plano, mientras que, por otra parte, a partir de las medidas de los desplazamientos en dirección perpendicular a la cara, se puede determinar el CET fuera del plano.

4.1 Determinación de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\alpha }}}_{\mathit{\boldsymbol{1}}}} y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\alpha }}}_{\mathit{\boldsymbol{2}}}} 4.1 Determinación de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\alpha }}} {\mathit{\boldsymbol{1}}}} y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\alpha }}} {\mathit{\boldsymbol{2}}}}

Representando las deformaciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}}
de la Tabla 1, se obtienen los resultados que se muestran en la Figura 5.

Dado que las probetas 0 y 90 tienen todas las láminas en la misma dirección (y están orientadas perpendicularmente las fibras de una probeta respecto de las de la otra) se observa que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}^{0}\cong {\epsilon }_{yy}^{90}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}^{0}\cong {\epsilon }_{xx}^{90}}
(el superíndice indica la probeta en que se realizan las medidas).

En la probeta QI, la disposición de las láminas hace que las propiedades en direcciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
sean coincidentes, por lo que debe observarse Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}^{QI}\cong {\epsilon }_{yy}^{QI}}

.

Por consiguiente, ajustando por una recta las medidas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}^{0}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}^{90}}
en función de la temperatura, se consigue el valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{1}}

. Mediante un ajuste de mínimos cuadrados (obligando a que la recta pase por el origen de coordenadas) se obtiene Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{1}=36.4\, \mu \epsilon ^oC}

(que coincide con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{x}^{0}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}^{90}}

).

Análogamente, ajustando las medidas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{yy}^{0}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\epsilon }_{xx}^{90}}
se obtiene un resultado de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{1}=-0.7\, \mu \epsilon  ^oC}
(que coincide con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{x}^{90}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}^{0}}

).

Siguiendo el mismo procedimiento, para la probeta QI se obtiene un resultado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{x}^{QI}={\alpha }_{y}^{QI}=1.4\, \mu \epsilon ^0C} .

Con objeto de dar una idea de la precisión de las medidas, se recogen en la Tabla 3 los ajustes llevados a cabo individualmente para cada probeta. Como puede observarse, los resultados que deben ser coincidentes muestran valores muy similares.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mu \epsilon   ^0C]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mu \epsilon   ^0C]}

4.2 Determinación de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{\alpha }}}_{\mathit{\boldsymbol{3}}}}

A la vista de (1), es evidente que el giro unitario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta /\beta}

 sufrido por la cara de la probeta se relaciona linealmente con el incremento de temperatura.

Representando las medidas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta /\beta}

 (calculadas a partir del giro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta \beta}
 de la Tabla 2 y considerando que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta \cong 90\mbox{°}}

), se obtienen los resultados que se muestran en la Figura 6. Como puede observarse, los resultados se ajustan razonablemente bien con el comportamiento lineal expresado por la ecuación (1), especialmente para las temperaturas más elevadas.

Mediante un ajuste de mínimos cuadrados (obligando a que la recta pase por el origen de coordenadas) se obtienen los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( {\alpha }_{z}-\right. } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\alpha }_{y}\right)

 mostrados para cada una de las probetas en la Tabla 4.

En la Tabla 4 se han recogido también, a modo de resumen, los valores obtenidos en el apartado anterior para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{x}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{y}}

, junto con el valor resultante para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\alpha }_{z}} .

 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mu \epsilon   ^0C]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mu \epsilon   ^0C]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mu \epsilon   ^0C]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mu \epsilon   ^0C]}

5 Conclusiones

Se ha llevado a cabo una medida de los coeficientes de expansión térmica (CET) dentro y fuera del plano en probetas en forma de L con diferentes orientaciones.

Para ello se ha empleado la técnica de videocorrelación, a partir de la cual se han determinado los desplazamientos que sufre una de las caras de la probeta, al ser sometida a un incremento de temperatura dentro de un horno.

A partir de los desplazamientos dentro del plano de la cara, el sistema de análisis de las imágenes obtenidas antes y después del calentamiento permite determinar las componentes del tensor de deformaciones. Las medidas obtenidas proporcionan valores relativamente uniformes de las deformaciones en las caras, los cuales permiten determinar los CET dentro del plano.

Al ser dos de las probetas fabricadas con todas las láminas en idéntica orientación, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ {0\mbox{°}}_{8}\right]}

 y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{90\mbox{°}}_{8}]}

, los CET dentro del plano coinciden con los CET del material en las direcciones principales de ortotropía dentro del plano.

Los resultados muestran que la técnica empleada permite determinar los CET dentro del plano con gran precisión, dado que los resultados obtenidos con ambas probetas son muy similares.

En la probeta de secuencia de apilado cuasi-isótropa, los resultados obtenidos para los CET dentro del plano son también muy similares. No obstante, el error obtenido en esta probeta es algo mayor, lo cual puede estar motivado por ligeros errores en la alineación de las capas a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 45\mbox{°}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -45\mbox{°}}

. En cualquier caso, los resultados obtenidos son satisfactorios.

Por último, el análisis de las medidas de los desplazamientos en la dirección perpendicular a la cara de la probeta permite determinar el ángulo girado por la misma y, por tanto, la discrepancia entre el CET en dirección del espesor y en la dirección dentro del plano afectada por la curvatura.

A partir de esta discrepancia y las medidas efectuadas dentro del plano, puede obtenerse el valor del CET en la dirección del espesor.

Los resultados obtenidos para el valor del CET en la dirección del espesor, muestran, por un lado, unos valores significativamente más elevados que el CET del material en la dirección dentro del plano perpendicular a las fibras y, por otro, una dispersión considerable entre unas probetas y otras.

Es preciso tener en cuenta que, mientras que las medidas de las deformaciones dentro del plano proporcionan los valores de los CET dentro del plano en la cara plana de la probeta en la que se realizan las medidas, las medidas de los desplazamientos fuera del plano proporcionan (indirectamente) la diferencia entre los valores del CET en la dirección del espesor y el CET afectado por la curvatura en la zona del codo de la probeta. En dicha zona, es preciso doblar las fibras para conseguir formar el codo, lo que provoca que estas puedan desviarse significativamente de su orientación teórica, afectando por tanto a las propiedades en dicha zona.

Por consiguiente, la falta de repetitividad en los resultados proporcionados por las distintas probetas no se considera asociada a un defecto del proceso de medición sino a una falta de homogeneidad en las probetas (defecto que aparece típicamente en componentes industriales y es inherente al procedimiento de fabricación), que hace que las propiedades sean, en realidad, diferentes en la zona del codo y en el resto de la probeta

Agradecimientos

El presente trabajo ha sido parcialmente financiado por el Ministerio de Economía y Competitividad (Proyecto MAT2015-71309-P) y el Ministerio de Economía, Industria y Competitividad y el Fondo Europeo de Desarrollo Regional (Proyecto UNSE15-CE-3581, MINECO/FEDER, UE).

Referencias

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[3] D.W.Radford, R.J.Diefendorf, Journal of Reinforced Plastics and Composites, 12, pág. 58-75 (1993).
https://doi.org/10.1177/073168449301200104

[4] C.Dudescu, J Naumann, M.Stockmann, S.Nebel, Strain, 42, pág. 197-205 (2006).
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[6] AS4/8552. Hexcel product data sheet (2018).
http://www.hexcel.com/user_area/content_media/raw/
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